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All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Residuals-based distributionally robust optimization with covariate information

We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of regression setups and DRO ambiguity sets. We investigate asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through numerical experiments, we validate our theoretical results, study the effectiveness of our approaches for sizing ambiguity sets, and illustrate the benefits of our DRO formulations in the limited data regime even when the prediction model is misspecified.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Revealing Decision Conservativeness Through Inverse Distributionally Robust Optimization

This paper introduces Inverse Distributionally Robust Optimization (I-DRO) as a method to infer the conservativeness level of a decision-maker, represented by the size of a Wasserstein metric-based ambiguity set, from the optimal decisions made using Forward Distributionally Robust Optimization (F-DRO). By leveraging the Karush-Kuhn-Tucker (KKT) conditions of the convex F-DRO model, we formulate I-DRO as a bi-linear program, which can be solved using off-the-shelf optimization solvers. Additionally, this formulation exhibits several advantageous properties. We demonstrate that I-DRO not only guarantees the existence and uniqueness of an optimal solution but also establishes the necessary and sufficient conditions for this optimal solution to accurately match the actual conservativeness level in F-DRO. Furthermore, we identify three extreme scenarios that may impact I-DRO effectiveness. Our case study applies F-DRO for power system scheduling under uncertainty and employs I-DRO to recover the conservativeness level of system operators. Numerical experiments based on an IEEE 5-bus system and a realistic NYISO 11-zone system demonstrate I-DRO performance in both normal and extreme scenarios. An extended version of this paper with additional analyses is available at li2024revealing.

distributionally robust optimization↗
DOE OSTI2022

A multistage distributionally robust optimization approach to water allocation under climate uncertainty

This paper investigates a Multistage Distributionally Robust Optimization (MDRO) approach to water allocation under climate uncertainty. The MDRO is formed by creating sets of conditional distributions (called conditional ambiguity sets) on a finite scenario tree. The distributions in the conditional ambiguity sets remain close to a nominal conditional distribution according a ø-divergence (e.g., Kullback-Leibler divergence, Hellinger distance, Burg entropy, etc.). Here, the paper discusses a decomposition algorithm to solve the resulting MDRO with ø-divergences, which uses the dual formulation and solves only linear subproblems instead of convex ones. Some properties of the algorithm such as generating feasible policies and valid upper/lower bounds are established. The paper then applies the modeling and solution techniques to allocate water in a rapidly-developing area of Tucson, Arizona. Tucson, like many arid and semi-arid regions around the world, faces considerable uncertainty in its ability to provide water for its citizens in the future. The primary sources of uncertainty in the Tucson region include (1) unpredictable population growth, (2) the availability of water from the Colorado River, and (3) the effects of climate variability on water consumption. This paper integrates forecasts for all these sources of uncertainty into a single optimization model for robust and sustainable water allocation. Then, it uses this model to analyze the value of constructing additional treatment facilities to reduce future water shortages. The results indicate that the MDRO approach can be very valuable for water managers by providing insights to minimize their risks and help them plan for the future.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2022

Residuals-based distributionally robust optimization with covariate information

We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of regression setups and DRO ambiguity sets. We investigate asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through numerical experiments, we validate our theoretical results, study the effectiveness of our approaches for sizing ambiguity sets, and illustrate the benefits of our DRO formulations in the limited data regime even when the prediction model is misspecified.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

A Distributionally Robust Optimization Framework for Stochastic Assessment of Power System Flexibility in Economic Dispatch

Given the complexity of power systems, particularly the high-dimensional variability of net loads, accurately depicting the entire operational range of net loads poses a challenge. To address this, recent methodologies have sought to gauge the maximum range of net load uncertainty across all buses. In this paper, we consider the stochastic nature of the net load and introduce a distributionally robust optimization framework that assesses system flexibility stochastically, accommodating a minimal extent of system violations. We verify the proposed method by solving the flexibility of the economic dispatch problem on four distinct IEEE standard test systems. Compared to traditional deterministic flexibility evaluations, our approach consistently yields less conservative flexibility outcomes.

distributionally robust optimization↗
DOE OSTI2022

Mitigating the Impacts of Uncertain Geomagnetic Disturbances on Electric Grids: A Distributionally Robust Optimization Approach

Severe geomagnetic disturbances (GMDs) increase the magnitude of the electric field on the Earth’s surface (E-field) and drive geomagnetically-induced currents (GICs) along the transmission lines in electric grids. These additional currents can pose severe risks, such as current distortions, transformer saturation and increased reactive power losses, each of which can lead to system unreliability. Today several mitigation actions (e.g., changing grid topology) exist that can reduce the harmful GIC effects on the grids. Making such decisions can be challenging, however, because the magnitude and direction of the E-field are uncertain and non-stationary. In this paper, we model uncertain E-fields using the distributionally robust optimization (DRO) approach that determines optimal transmission grid operations such that the worst-case expectation of the system cost is minimized. We also capture the effect of GICs on the nonlinear AC power flow equations. For solution approaches, we develop an accelerated column-and-constraint generation (CCG) algorithm by exploiting a special structure of the support set of uncertain parameters representing the E-field. Extensive numerical experiments based on “epri-21” and “uiuc-150” systems, designed for GMD studies, demonstrate (i) the computational performance of the accelerated CCG algorithm, (ii) the superior performance of distributionally robust grid operations that satisfy nonlinear, nonconvex AC power flow equations and GIC constraints, in comparison with standard stochastic programming-based methods during the out-of-sample testing.

42 ENGINEERING↗
DOE OSTI2024

Improved microgrid resiliency through distributionally robust optimization under a policy-mode framework

Critical energy infrastructure are constantly under stress due to the ever increasing disruptions caused by wildfires, hurricanes, other weather related extreme events and cyber-attacks. Hence it becomes important to make critical infrastructure resilient to threats from such cyber-physical events. However, such events are hard to predict and numerous in nature and type and it becomes infeasible to make a system resilient to every possible such cyber-physical event. Such an approach can make the system operation overly conservative and impractical to operate. Furthermore, distributions of such events are hard to predict and historical data available on such events can be very sparse, making the problem even harder to solve. To deal with these issues, in this paper we present a policy-mode framework that enumerates and predicts the probability of various cyber-physical events and then a distributionally robust optimization (DRO) formulation that is robust to the sparsity of the available historical data. The proposed algorithm is illustrated on an islanded microgrid example: a modified IEEE 123-node feeder with distributed energy resources (DERs) and energy storage. Simulations are carried to validate the resiliency metrics under the sampled disruption events.

Nazir, Mohammad Nawaf↗
DOE OSTI2025

Learning with Adaptive Conservativeness for Distributionally Robust Optimization: Incentive Design for Voltage Regulation

Information asymmetry between the Distribution System Operator (DSO) and Distributed Energy Resource Aggregators (DERAs) obstructs designing effective incentives for voltage regulation. To capture this effect, we employ a Stackelberg game-theoretic framework, where the DSO seeks to overcome the information asymmetry and refine its incentive strategies by learning from DERA behavior over multiple iterations. We introduce a model-based online learning algorithm for the DSO, aimed at inferring the relationship between incentives and DERA responses. Given the uncertain nature of these responses, we also propose a distributionally robust incentive design model to control the probability of voltage regulation failure and then reformulate it into a convex problem. This model allows the DSO to periodically revise distribution assumptions on uncertain parameters in the decision model of the DERA. Finally, we present a gradient-based method that permits the DSO to adaptively modify its conservativeness level, measured by the size of a Wasserstein metric-based ambiguity set, according to historical voltage regulation performance. The effectiveness of our proposed method is demonstrated through numerical experiments.

adaptation models↗
DOE OSTI2024

Learning with Adaptive Conservativeness for Distributionally Robust Optimization: Incentive Design for Voltage Regulation: Preprint

Information asymmetry between the Distribution System Operator (DSO) and Distributed Energy Resource Aggregators (DERAs) obstructs designing effective incentives for voltage regulation. To capture this effect, we employ a Stackelberg game-theoretic framework, where the DSO seeks to overcome the information asymmetry and refine its incentive strategies by learning from DERA behavior over multiple iterations. We introduce a model-based online learning algorithm for the DSO, aimed at inferring the relationship between incentives and DERA responses. Given the uncertain nature of these responses, we also propose a distributionally robust incentive design model to control the probability of voltage regulation failure and then reformulate it into a convex problem. This model allows the DSO to periodically revise distribution assumptions on uncertain parameters in the decision model of the DERA. Finally, we present a gradient-based method that permits the DSO to adaptively modify its conservativeness level, measured by the size of a Wasserstein metric-based ambiguity set, according to historical voltage regulation performance. The effectiveness of our proposed method is demonstrated through numerical experiments.

distribution system operator↗
DOE OSTI2024

Distributionally Robust Bilevel Optimization Model for Distribution Network With Demand Response Under Uncertain Renewables Using Wasserstein Metrics

Here, we consider a distribution network integrating demand response (DR) participants in the presence of uncertain renewable suppliers and outdoor temperatures. A bilevel optimization model is proposed to capture the intricate dynamics between price-incentivized DR participants and distribution system operations, including energy procurement and active/reactive power flows. The model is formulated as a distributional robust bilevel optimization using Wasserstein metrics. We show favorable data-driven properties including out-of-sample guarantee and asymptotic consistency. Furthermore, we present a tractable mixed-integer linear programming reformulation and characterize the worst-case distribution. Computational experiments are conducted on a modified 33-bus system. Our findings underscore the efficacy of the pricing strategies derived from the proposed bilevel optimization model. These strategies not only effectively manage DR participants' behavior but also bring equity considerations among households with various characteristics to light. The results contribute to a deeper understanding of the interplay between distribution system operators and DR participants.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

Distributionally Robust Decision Making Leveraging Conditional Distributions

Distributionally robust optimization (DRO) is a powerful tool for decision making under uncertainty. It is particularly appealing because of its ability to leverage existing data. However, many practical problems call for decision-making with some auxiliary information, and DRO in the context of conditional distributions is not straightforward. We propose a conditional kernel distributionally robust optimization (CKDRO) method that enables robust decision making under conditional distributions through kernel DRO and the conditional mean operator in the reproducing kernel Hilbert space (RKHS). In particular, we consider problems where there is a correlation between the unknown variable y and an auxiliary observable variable x. Given past data of the two variables and a queried auxiliary variable, CKDRO represents the conditional distribution P(y|x) as the conditional mean operator in the RKHS space and quantifies the ambiguity set in the RKHS as well, which depends on the size of the dataset as well as the query point. To justify the use of RKHS, we demonstrate that the ambiguity set defined in RKHS can be viewed as a ball under a metric that is similar to the Wasserstein metric. The DRO is then dualized and solved via a finite dimensional convex program. The proposed CKDRO approach is applied to a generation scheduling problem and shows that the result of CKDRO is superior to common benchmarks in terms of quality and robustness.

Chen, Yuxiao↗
DOE OSTI2026

Dynamic Transmission Line Switching Amid Wildfire-Prone Weather Under Decision-Dependent Uncertainty

During dry and windy seasons, environmental conditions significantly increase the risk of wildfires, exposing power grids to disruptions caused by transmission line failures. Wildfire propagation exacerbates grid vulnerability, potentially leading to prolonged power outages. To address this challenge, we propose a multistage optimization model that dynamically adjusts transmission grid topology in response to wildfire propagation, aiming to develop an optimal response policy. By accounting for decision-dependent uncertainty, where line survival probabilities depend on usage, we employ distributionally robust optimization to model uncertainty in line survival distributions. We adapt the stochastic nested decomposition algorithm and derive a deterministic upper bound for its finite convergence. To enhance computational efficiency, we exploit the Lagrangian dual problem structure for a faster generation of Lagrangian cuts. Using realistic data from the California transmission grid, we demonstrate the superior performance of dynamic response policies against two-stage alternatives through a comprehensive case study. In addition, after solving the multistage formulation, we construct easy-to-implement policies that significantly reduce computational burden while maintaining good performance in real-time deployment. History: Accepted by Russell Bent, Area Editor for Network Optimization: Algorithms and Applications. Funding: This work was supported by the U.S. Department of Energy, Office of Electricity [Grant DE-AC02-05CH11231]. The work of R. Jiang was supported in part by the U.S. National Science Foundation, Division of Electrical, Communications and Cyber Systems [Grant ECCS-1845980] and the U.S. Air Force Office of Scientific Research [Grant FA9550-23-1-0323]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2025.1210 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2025.1210 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Estrada-Garcia, Juan-Alberto↗
DOE OSTI2022

Incorporate day-ahead robustness and real-time incentives for electricity market design

In this paper, we propose a two-stage electricity market framework to explore the participation of distributed energy resources (DERs) in a day-ahead (DA) market and a real-time (RT) market. The objective is to determine the optimal bidding strategies of the aggregated DERs in the DA market and generate online incentive signals for DER-owners to optimize the social-welfare taking into account network operational constraints. Distributionally robust optimization is used to explicitly incorporate data-based statistical information of renewable forecasts into the supply/demand decisions in the DA market. We evaluate the conservativeness of bidding strategies distinguished by different risk aversion settings. In the RT market, a bi-level time-varying optimization problem is proposed to design the online incentive signals to tradeoff the RT imbalance penalty for distribution system operators (DSOs) and the costs of individual DER-owners. This enables tracking their optimal dispatch to provide fast balancing services, in the presence of time-varying network states while satisfying the voltage regulation requirement. Simulation results on both DA wholesale market and RT balancing market demonstrate the necessity of this two-stage design, and its robustness to uncertainties, the performance of convergence, the tracking ability and the feasibility of the resulting network operations.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

Robust Restoration From Cyber-Physical Attacks in Active Distribution Grids With Grid-Edge IBRs

The inverter-based resources (IBRs) have enabled the integration of renewable energy at the grid edge with enhanced control capabilities to support the reliable operation of power grids. Different control frameworks, such as hierarchical or distributed architecture, have been proposed with the expansion of cyber networks for real-time monitoring and control. This evolution of critical infrastructure into cyber-physical systems also brings more vulnerabilities for the broadened attack surfaces, and significantly increases the possibility of physical system failures or outages caused by cyberattacks. Among tremendous efforts in the defense-in-depth approach, it remains challenging to provide prompt detection and accurate location of attack entry points or paths. Therefore, the prevailing restoration framework may struggle to fully consider the cyber-physical interdependence, successfully isolate the compromised cyber and physical components, and safely recover the systems without the potential risks leading to secondary outages. This paper is motivated to develop a cyber-physical restoration framework for distribution grids to recover from cyber attacks by harnessing grid-edge IBRs. The framework is first built on the operational guidelines of IBRs considering the compromised cyber layer. Then, an ambiguity set is established to represent the uncertainty of attack scenarios and their possibility levels. Next, a distributionally robust optimization model is developed to provide the optimal load restoration strategy across all scenarios. The effectiveness of the proposed model is demonstrated through various use cases on the modified IEEE 13-node and 123-node test systems. Finally, simulation results demonstrate the effectiveness and advancement of developed post-attack restoration strategies.

Cybersecurity↗
DOE OSTI2025

Enhancing the Survivability of Power Systems With Grid-Edge DERs Against DoS Attacks

Power system survivability, defined as the ability of a system to maintain steady-state functionality under varying operational conditions, reflects its resilience against disturbances. While existing research primarily focuses on physical-layer disturbances, the increasing prevalence of grid-edge DERs, which are primarily used for integrating renewable energy, has significantly expanded the cyber attack surface. As a result, operational disruptions caused by cyber threats are posing significant challenges to system survivability and cannot be overlooked. To fill this gap, we redefine system survivability to incorporate the cyber layer’s status and propose a Distributionally Robust Optimization (DRO) approach to enhance power system survivability against potential cyber-physical threats. In this paper, we first analyze the operational guidelines of systems with a high penetration of DERs under various cyber network conditions and redefine survivability in this context. Next, we focus on the most common cyber threat, Denial-of-Service (DoS) attacks, and develop a corresponding attack model. This model allows for the creation of a kernel-based ambiguity set that captures attack uncertainties using historical data. Finally, we transform the proposed DRO model as a tractable optimization problem, with its solution providing an optimal cyber redundancy plan to enhance system survivability in DoS attack scenarios. Simulation results on the IEEE 13-node and 123-node test feeders demonstrate the effectiveness of our proposed model in improving system survivability. This model can also be expanded to include other types of common attacks and serve as a comprehensive planning tool to improve overall cyber physical survival of the system.

cybersecurity↗
DOE OSTI2025

Optimization problems governed by systems of PDEs with uncertainties

This paper reviews current theoretical and numerical approaches to optimization problems governed by partial differential equations (PDEs) that depend on random variables or random fields. Such problems arise in many engineering, science, economics and societal decision-making tasks. This paper focuses on problems in which the governing PDEs are parametrized by the random variables/fields, and the decisions are made at the beginning and are not revised once uncertainty is revealed. Examples of such problems are presented to motivate the topic of this paper, and to illustrate the impact of different ways to model uncertainty in the formulations of the optimization problem and their impact on the solution. A linear–quadratic elliptic optimal control problem is used to provide a detailed discussion of the set-up for the risk-neutral optimization problem formulation, study the existence and characterization of its solution, and survey numerical methods for computing it. Different ways to model uncertainty in the PDE-constrained optimization problem are surveyed in an abstract setting, including risk measures, distributionally robust optimization formulations, probabilistic functions and chance constraints, and stochastic orders. Furthermore, approximation-based optimization approaches and stochastic methods for the solution of the large-scale PDE-constrained optimization problems under uncertainty are described. Some possible future research directions are outlined.

Heinkenschloss, Matthias [Rice Univ., Houston, TX ↗