The absence of formal performance guarantees in machine learning (ML) has limited its adoption for safety-critical power system applications, where confidence and interpretability are as vital as accuracy. In this work, we present a probabilistic guarantee for power flow learning and voltage risk estimation, derived through the framework of Gaussian Process (GP) regression. Specifically, we establish a bound on the expected estimation error that connects the GP’s predictive variance to confidence in voltage risk estimates, ensuring statistical equivalence with Monte Carlo-based ACPF risk quantification. To enhance model learnability in the low-data regime, we first design the Vertex-Degree Kernel (VDK), a topology-aware additive kernel that decomposes voltage-load interactions into local neighborhoods for efficient large-scale learning. Building on this, we introduce a network-swipe active learning (AL) algorithm that adaptively samples informative operating points and provides a principled stopping criterion without requiring out-of-sample validation. Together, these developments mitigate the principal bottleneck of ML-based power flow-its lack of guaranteed reliability-by combining data efficiency with analytical assurance. Empirical evaluations across IEEE 118-, 500-, and 1354-bus systems confirm that the proposed VDK-GP achieves mean absolute voltage errors below 1E-03 p.u., reproduces Monte Carlo-level voltage risk estimates with 15 ×  fewer ACPF computations, and achieves over 120 ×  reduction in evaluation time while conservatively bounding violation probabilities.

Reinforcement learning (RL) has shown great potential for designing voltage control policies, but their performance often degrades under changing system conditions such as topology reconfigurations and load variations. We introduce a topology-aware online policy optimization framework that leverages data-driven estimation of voltage-reactive power sensitivities to achieve efficient policy adaptation for medium-voltage radial distribution networks. Exploiting the sparsity of topology-switching events, where only a few lines change at a time, our method efficiently detects topology changes and identifies the affected lines and parameters, enabling fast and accurate sensitivity updates without recomputing the full sensitivity matrix. The estimated sensitivity is subsequently used for online policy optimization of a pre-trained neural-network-based RL controller. Simulations on both the IEEE 13-bus and SCE 56-bus systems demonstrate over 90% line identification accuracy, using only 15 data points. The proposed method also significantly improves voltage regulation performance compared with non-adaptive policies and adaptive policies that rely on regression-based online optimization methods for sensitivity estimation.

With the increasing penetration of intermittent inverter-based resources (IBRs) and flexible loads, identifying critical variables that affect the monitoring and control of stochastic power systems is essential for ensuring secure operation. This paper proposes a hierarchically spatio-temporal global sensitivity analysis framework for probabilistic transient stability assessment (PTSA), capable of quantifying the temporal and spatial sensitivity to correlated uncertain sources, such as IBRs and loads, for correlated generator rotor angle dynamics. To accelerate the computation of sensitivity indices, a computationally efficient surrogate model, i.e., a multi-output Gaussian process emulator (MOGPE), is further developed to replace the time-consuming time-domain simulator used in PTSA. Numerical results on the modified IEEE 9-bus system and the NPCC 140-bus system demonstrate the accuracy, efficiency, and scalability of the proposed method.

Large, spatially flexible electricity consumers such as data centers can reallocate demand across locations, influencing dispatch and prices in wholesale electricity markets. While flexible load is often assumed to improve system efficiency, this intuition typically relies on price-taking behavior. We study price-anticipatory spatial load shifting by modeling a large flexible consumer as a Stackelberg leader interacting with DC optimal power flow (DC-OPF) based market clearing. We show that decentralized, cost-minimizing load shifting need not align with system operating cost minimization, and that misalignment arises at boundaries between DC-OPF operating regimes, where small changes in load can induce discrete changes in marginal generators or congestion patterns. We evaluate strategic load shifting on the 73-bus RTS-GMLC test system, where findings indicate reductions in system operating cost in most hours, but misalignment in a subset of cases that are driven by redispatch at merit-order discontinuities. We find that these outcomes are primarily redistributive relative to a price-taking benchmark, reducing generator profits while lowering electricity procurement costs for both flexible and inflexible consumers, even in cases where total system operating costs increase.

The increasing integration of renewable energy sources (RESs) and distributed energy resources (DERs) has significantly heightened operational complexity and uncertainty in modern power systems. Concurrently, the widespread deployment of smart meters, phasor measurement units (PMUs) and other sensors has generated vast spatiotemporal data streams, enabling advanced data-driven analytics and decision-making in grid operations. In this context, Gaussian processes (GPs) have emerged as a powerful probabilistic framework, offering uncertainty quantification, non-parametric modeling, and predictive capabilities to enhance power system analysis and control. This paper presents a comprehensive review of GP techniques and their applications in power system operation and control. GP applications are reviewed across three key domains: GP-based modeling, risk assessment, and optimization and control. These areas serve as representative examples of how GPs can be utilized in power systems. Furthermore, critical challenges in GP applications are discussed, and potential research directions are outlined to facilitate future power system operations.

The Alternating Current Optimal Power Flow (ACOPF) problem is a core task in power system operations, aimed at determining cost-effective generation dispatch while satisfying physical and operational constraints. However, conventional ACOPF formulations rely on steady-state models and neglect generator dynamics, which can result in operating points that are economically optimal but dynamically unstable. This paper proposes a novel, data-driven approach to incorporate generator dynamics into the ACOPF using Gaussian Process (GP) models. Specifically, it introduces an exponential surrogate function to characterize the stability of solutions to the differential equations governing synchronous generator dynamics. The exponent, which indicates whether system trajectories decay (stable) or grow (unstable), is learned as a function of the bus voltage using GP regression. Crucially, the framework enables probabilistic stability assessment to be integrated directly into the optimization process. The resulting dynamics-aware ACOPF formulation identifies operating points that satisfy both operational safety and dynamic stability criteria. Numerical experiments on the IEEE 39-bus, 57-bus, and 118-bus systems demonstrate that, compared with existing data-driven approaches, the proposed method efficiently captures generator dynamics with limited training data, yielding more reliable and robust decisions across a wide range of operating conditions.

Enabling continued data-center growth under increasing grid stress motivates closer coordination between flexible computing demand and co-located battery energy storage systems (BESS) to improve site operations and provide grid services. This paper develops a robust co-optimization framework for day-ahead operation of data centers with co-located BESS under utility-imposed interconnection limits on peak load and ramping. The model jointly considers deadline-constrained computing workloads, managed through workload scheduling and dynamic voltage and frequency scaling (DVFS), together with degradation-aware BESS dispatch to enable cost optimization and participation in ancillary-service markets. Case studies based on real-world market and workload data show that the proposed framework yields feasible day-ahead schedules across a range of operating conditions, with substantially larger benefits when interconnection constraints become binding. Under baseline conditions, BESS value is derived from both ancillary-service participation and improved workload and energy management. Under stressed peak-load and ramping limits, however, the daily value of BESS increases by a factor of two or more, driven primarily by BESS actions to reduce the potential incompletion in the schedulable workload while complying with interconnection constraints. Under tight peak-load caps, workload composition also matters where a higher share of non-schedulable jobs can increase operating cost by more than 25% relative to more flexible workload mixes. Additionally, DVFS studies further show that processor-level control is a material flexibility lever under tight load limits. These results demonstrate that coordinated compute-storage flexibility can materially expand the operational headroom and grid value of data centers, especially under increasingly scarce grid capacity.

Controlling the dispersion of a subset of decision variables in an optimization problem is crucial for enforcing fairness or load-balancing across a wide range of applications. Building on the well-known equivalence of finite-dimensional norms, the article develops a family of parameterized convex models that regulate the dispersion of a vector of decision-variable values through its coefficient of variation. Each model has a single parameter taking values in the interval [0, 1]. When the parameter is set to zero, the model imposes only a trivial constraint on the optimization problem; when set to one, it enforces equality of all the decision variables. As the parameter varies, the coefficient of variation is provably bounded above by a monotonic function of that parameter. The article also presents theoretical results relating the space of feasible solutions across all models. Finally, it compares the models’ solution quality on a variant of the assignment problem that regulates the dispersion in the assignment costs.

This paper proposes a novel learning-based approach to approximately solve instances of mixed-integer optimization problems. These problems are computationally challenging, as they require jointly determining discrete and continuous decisions while satisfying complex combinatorial constraints. The proposed method relies on a graph-based generative diffusion model that learns the discrete component of mixed-integer optimization problems while integrating a training-free feasibility projection operator directly into the reverse diffusion process to steer intermediate samples toward the feasible set throughout generation. Once the discrete decisions are generated, the remaining optimization reduces to a continuous problem that can be solved efficiently (relative to the original problem) using existing numerical methods. The resulting framework named Constrained Graph Diffusion (CGD), is problem-agnostic and can accommodate a broad class of mixed-integer optimization problems through suitable projection operators. We evaluate CGD on optimal transmission switching for ACOPF and discrete portfolio optimization, demonstrating substantial improvements in feasibility and solution quality over learning-based baselines while achieving speedups of up to 425× over state-of-the-art numerical solvers for MINLPs.

Dynamic operating envelopes (DOEs) provide a systematic framework to integrate the flexibility of distribution grid resources while safeguarding network limits such as line ratings and voltage bounds. However, the flexibility derived from individual DOEs is often restricted and conservative, especially when some resources can coordinate via communication with an aggregator. This paper presents a convex, geometry-aware framework for constructing DOE for distribution grid customers under partial coordination, with coordinated customers modeled through polytopal flexibility sets and non-coordinated customers through hyperrectangles. The framework additionally incorporates fairness constraints for export and import headroom allocated to the customers within the DOE design. To account for forecast uncertainty in inelastic injections, the DOE design is extended to a robust formulation for bounded uncertainty sets. Case studies on two European three-phase low-voltage feeders, a widely used test feeder and a large-scale (3589)-bus system, show that the proposed DOE construction expands aggregate flexibility while maintaining network feasibility, fairness, and robustness to forecast uncertainty. Coordinating 30

We learn optimal instance-specific heuristics for the global minimization of nonconvex quadratically constrained quadratic programs (QCQPs). Specifically, we consider partitioning-based convex mixed-integer programming relaxations for nonconvex QCQPs and propose the novel problem of strong partitioning to optimally partition variable domains without sacrificing global optimality. Because solving this max-min strong partitioning problem exactly can be very challenging, we design a local optimization method that leverages generalized gradients of the value function of its inner-minimization problem. However, even solving the strong partitioning problem to local optimality can be time consuming. To address this, we propose a simple and practical machine learning (ML) approximation for homogeneous families of QCQPs. Motivated by practical applications, we conduct a detailed computational study using the open-source global solver Alpine to evaluate the effectiveness of our ML approximation in accelerating the repeated solution of homogeneous QCQPs with fixed structure. Our study considers randomly generated QCQP families, including instances of the pooling problem, that are benchmarked using state-of-the-art global optimization software. Numerical experiments demonstrate that our ML approximation of strong partitioning reduces Alpine's solution time by a factor of 2-4.5 on average, with maximum reduction factors ranging from 10 to 200 across these QCQP families.

Constrained optimization problems arise in various engineering systems such as inventory management and power grids. Standard deep neural network (DNN) based machine learning proxies are ineffective in practical settings where labeled data is scarce and training times are limited. We propose a semi-supervised Bayesian Neural Networks (BNNs) based optimization proxy for this complex regime, wherein training commences in a sandwiched fashion, alternating between a supervised learning step for minimizing cost, and an unsupervised learning step for enforcing constraint feasibility. We show that the proposed semi-supervised BNN outperforms DNN architectures on important non-convex constrained optimization problems from energy network operations, achieving up to a tenfold reduction in expected maximum equality gap and halving the inequality gaps. Further, the BNN's ability to provide posterior samples is leveraged to construct practically meaningful probabilistic confidence bounds on performance using a limited validation data, unlike prior methods.

Electricity distribution companies deploy battery storage to defer grid upgrades by reducing peak demand. In deregulated jurisdictions, such storage often sits idle because regulatory constraints bar participation in electricity markets. Here, we develop an optimization framework that, to our knowledge, provides the first formal model of market participation constraints within storage investment and operation planning. Applying the framework to a Massachusetts case study, we find that market participation could deliver similar savings as peak demand reduction. Under current conditions, market participation does not increase storage investment, but at very low storage costs, could incentivize deployment beyond local distribution needs. This might run contrary to the separation of distribution from generation in deregulated markets. Our framework can identify investment levels appropriate for local distribution needs.

The software and data in this repository are a snapshot of the software and data that were used in the research reported in the paper by R. Kannan, H. Nagarajan, and D. Deka.

This paper provides a conceptual framework for enhancing the efficiency, reliability, and security of electricity transmission and distribution network (DN) operations by leveraging the flexibility of the DN to solve operational issues in transmission networks. The framework ensures that the aggregation and disaggregation adhere to the network constraints of DNs and the operational limitations of flexible processes. Grid flexibility is crucial for balancing supply and demand amid renewable energy uncertainties, but quantifying the flexibility of distributed resources (EVs, heat pumps, industrial loads) is complex due to their heterogeneous nature and constraints. Aggregators must account for the diversity, operational limits, and uncertainty of these sources to ensure reliable flexibility aggregation. Similarly, disaggregating flexibility necessitates precise locational and temporal activation to meet transmission grid needs while avoiding disruption in DN operations. Coordination between transmission and distribution system operators is critical to maintaining system stability. The paper proposes a conceptual framework to enable efficient aggregation and disaggregation of downstream flexibility resources for upstream transmission needs, ensuring the healthy operation of both transmission and DNs.

Complex networked systems driven by latent inputs are common in fields like neuroscience, finance, and engineering. A key inference problem here is to learn edge connectivity from node outputs (potentials). We focus on systems governed by steady-state linear conservation laws: $X_{t} = {L^{\ast }}Y_{t}$, where $X_{t}, Y_{t} \in \mathbb {R}^{p}$ denote inputs and potentials, respectively, and the sparsity pattern of the $p \times p$ Laplacian $L^{\ast }$ encodes the edge structure. Assuming $X_{t}$ to be a wide-sense stationary stochastic process with a known spectral density matrix, we learn the support of $L^{\ast }$ from temporally correlated samples of $Y_{t}$ via an $\ell _{1}$-regularized Whittle's maximum likelihood estimator (MLE). The regularization is particularly useful for learning large-scale networks in the high-dimensional setting where the network size $p$ significantly exceeds the number of samples $n$. We show that the MLE problem is strictly convex, admitting a unique solution. Under a novel mutual incoherence condition and certain sufficient conditions on $(n, p, d)$, we show that the ML estimate recovers the sparsity pattern of $L^\ast$ with high probability, where $d$ is the maximum degree of the graph underlying $L^{\ast }$. We provide recovery guarantees for $L^\ast$ in element-wise maximum, Frobenius, and operator norms. Finally, we complement our theoretical results with several simulation studies on synthetic and benchmark datasets, including engineered systems (power and water networks), and real-world datasets from neural systems (such as the human brain).

The adversarial worst-case load shedding (AWLS) problem is pivotal for identifying critical contingencies under line outages. It is naturally cast as a bilevel program: the upper level simulates an attacker determining worst-case line failures, and the lower level corresponds to the defender's generator redispatch operations. Conventional techniques using optimality conditions render the bilevel, mixed-integer formulation computationally prohibitive due to the combinatorial number of topologies and the nonconvexity of AC power flow constraints. To address these challenges, we develop a novel single-level optimal value-function (OVF) reformulation and further leverage a data-driven neural network (NN) surrogate of the follower's optimal value. To ensure physical realizability, we embed the trained surrogate in a physics-constrained NN (PCNN) formulation that couples the OVF inequality with (relaxed) AC feasibility, yielding a mixed-integer convex model amenable to off-the-shelf solvers. To achieve scalability, we learn a sparse, area-partitioned NN via spectral clustering; the resulting block-sparse architecture scales essentially linearly with system size while preserving accuracy. Notably, our approach produces near-optimal worst-case failures and generalizes across loading conditions and unseen topologies, enabling rapid online recomputation. Numerical experiments on the IEEE 14- and 118-bus systems demonstrate the method's scalability and solution quality for large-scale contingency analysis, with an average optimality gap of 5.8

Learning-based approaches are increasingly leveraged to manage and coordinate the operation of grid-edge resources in active power distribution networks. Among these, model-based techniques stand out for their superior data efficiency and robustness compared to model-free methods. However, effective model learning requires a learning-based approximator for the underlying power flow model. This study extends existing work by introducing a data-driven power flow method based on Gaussian Processes (GPs) to approximate the multiphase power flow model, by mapping net load injections to nodal voltages. Simulation results using the IEEE 123-bus and 8500-node distribution test feeders demonstrate that the trained GP model can reliably predict the nonlinear power flow solutions with minimal training data. We also conduct a comparative analysis of the training efficiency and testing performance of the proposed GP-based power flow approximator against a deep neural network-based approximator, highlighting the advantages of our data-efficient approach. Results over realistic operating conditions show that despite an 85% reduction in the training sample size (corresponding to a 92.8% improvement in training time), GP models produce a 99.9% relative reduction in mean absolute error compared to the baselines of deep neural networks.

The article develops a parametric model of fairness called "ε-fairness" that can be represented using a single second-order cone constraint and incorporated into existing decision-making problem formulations without impacting the complexity of solution techniques. We develop the model from the fundamental result of finite-dimensional norm equivalence in linear algebra and show that this model has a closed-form relationship to an existing metric for measuring fairness widely used in the literature. Finally, a simple case study on the optimal operation of a damaged power transmission network illustrates its effectiveness.

This work presents an efficient data-driven method to construct probabilistic voltage envelopes (PVE) using power flow learning in grids with network contingencies. First, a network-aware Gaussian process (GP) termed Vertex-Degree Kernel (VDK-GP), developed in prior work, is used to estimate voltage-power functions for a few network configurations. The paper introduces a novel multi-task vertex degree kernel (MT-VDK) that amalgamates the learned VDK-GPs to determine power flows for unseen networks, with a significant reduction in the computational complexity and hyperparameter requirements compared to alternate approaches. Simulations on the IEEE 30-Bus network demonstrate the retention and transfer of power flow knowledge in both N-1 and N-2 contingency scenarios. The MT-VDK-GP approach achieves over 50% reduction in mean prediction error for novel N-1 contingency network configurations in low training data regimes (50-250 samples) over VDK-GP. Additionally, MT-VDK-GP outperforms a hyper-parameter based transfer learning approach in over 75% of N-2 contingency network structures, even without historical N-2 outage data. The proposed method demonstrates the ability to achieve PVEs using sixteen times fewer power flow solutions compared to Monte-Carlo sampling-based methods.