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All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Piecewise polyhedral formulations for a multilinear term

Herein, we present a mixed-integer linear programming (MILP) formulation of a piecewise, polyhedral relaxation (PPR) of a multilinear term using its convex-hull representation. Based on the PPR’s solution, we also present a MILP formulation whose solutions are feasible for nonconvex, multilinear equations. We then present computational results showing the effectiveness of proposed formulations on standard benchmark nonlinear programs (NLPs) with multilinear terms and compare with a traditional formulation that is built using recursive bilinear groupings of multilinear terms.

42 ENGINEERING↗
DOE OSTI2026

Stochastic Microgrid Scheduling With Chance‐Constrained Resilience Consideration

Traditionally, it is assumed that microgrids transition seamlessly from grid‐connected operation to islanded mode in the event of sudden main grid outages. In reality, the islanding process, especially unintentional islanding, is rarely seamless. Instead, it is subject to voltage and frequency fluctuations caused by the instantaneous disconnection of the point of common coupling (PCC) switch, variations in loads and renewable generation output and even the protection tripping of distributed energy resources (DERs). To mitigate these fluctuations and facilitate a smooth islanding process, we propose a stochastic microgrid scheduling model that incorporates chance‐constrained resilience measures. Specifically, the resilience measure is defined as the probability of successful islanding (PSI), that is, the probability that a microgrid can mitigate the generation‐demand imbalance caused by the disconnection of the PCC switch, variations in load and renewable generation and DER tripping. This measure is modelled using chance constraints. Unlike existing reliability and resilience indices, which typically neglect the possibility of microgrid/DER failure under extreme events and assume their survival while primarily focussing on reducing impact duration or magnitude, the proposed PSI‐based framework explicitly addresses microgrid and DER survival during the islanding transition. The formulated nonlinear chance constraints are approximated using a multiinterval approach and equivalently represented as a mixed‐integer linear programming (MILP) formulation. Case study results validate the proposed method, showing that the PSI estimation error is reduced to less than 8%, compared to approximately 28% with existing methods. Various sensitivity analyses on the DER tripping rate and PSI settings were performed to validate the robustness of the proposed method. In particular, the necessity of accounting for DER tripping in the PSI calculation was demonstrated.

chance constrained optimization↗
DOE OSTI2025

An interregional optimization approach for time series aggregation in continent-scale electricity system models

Modeling electric power systems with high shares of weather-dependent resources requires tradeoffs between temporal, spatial, and operational resolution. Many studies perform time series aggregation using clustering algorithms to reduce the temporal dimension, but when modeling continent-scale electricity systems that are large enough to contain multiple independent weather systems, this approach requires large numbers of representative periods to minimize errors in regional wind and solar capacity factors. Here, a new optimization-based approach for representative period selection and weighting is introduced that minimizes regional errors in average renewable capacity factors and electricity demand. The method delivers higher regional fidelity with fewer representative periods than alternative clustering methods when applied to wind, solar, and demand profiles for the contiguous United States. When representative periods are selected from multiple weather years, the optimized method reproduces regional averages with lower error than a complete 365-day time series from any single weather year. The method identifies only representative (as opposed to outlying) periods but can be combined with an iterative "stress period" identification approach to guide efficient decision-making considering both average and high-risk weather conditions.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

Security-Constrained Unit Commitment for Electricity Market: Modeling, Solution Methods, and Future Challenges

This paper summarizes the technical activities of the IEEE Task Force on Solving Large Scale Optimization Problems in Electricity Market and Power System Applications. This Task Force was established by the IEEE Technology and Innovation Subcommittee to first review the state-of-the-art of the security-constrained unit commitment (SCUC) business model, its mathematical formulation, and solution techniques in solving electricity market clearing problems. The Task Force then investigated the emerging challenges of future market clearing problems and presented efforts in building benchmark mathematical and business models.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2025

Optimizing ensemble NV − spin properties of fluorescent diamond microparticles by systematic low pressure high temperature annealing

Low pressure high temperature annealing is a means for driving nitrogen and defect diffusion in diamond to reduce internal lattice damage without the need for technically complicated high-pressure cells. Herein, we perform a systematic time (5, 15, and 30 min) and temperature (1200 °C–1800 °C) study of effects of low-pressure high temperature annealing on photoluminescence, spin concentrations, and spin relaxation properties of NV centers in ca. 3 μm synthetic type 1b diamond particles. Annealing in the temperature range of ca. 1400 °C–1700 °C for even 5 min leads to a higher optically detected magnetic resonance contrast as compared to standard annealing at 900 °C for 2 h. Particles annealed at 1700 °C for 5 min exhibit a contrast close to about 13% as compared to about 9% for those annealed at 900 °C for 2 h. A reduction in the zero-field splitting strain parameter from E ≈ 4.5 MHz to ≈ 2.5 MHz and spectral linewidth from Δν ≈ 7 MHz to ≈ 4 MHz are observed even after 5 min annealing at 1700 °C. Improvements in these spectral parameters resulted in a roughly 2-fold reduction in the noise level of temperature monitoring experiment utilizing an ensemble of NV centers in the particles. Annealing in the temperature range of 1600 °C for 15 or 30 min or 1700 °C for 5 min resulted in NV T 1 relaxation times approaching ca. 5 ms typically observed for bulk diamond. Quantitative electron paramagnetic resonance (EPR) allowed for estimations of thermal activation energies of paramagnetic center annihilation. Monitoring the primary defect concentration (P1 and other defects with half integer spins) and utilizing second order kinetic modeling, an activation energy of 3.63 ± 0.28 eV was estimated. Alternatively, using the NV half field EPR signal and first order kinetic modeling, a similar activation energy 3.89 ± 0.29 eV was estimated.

NV centers↗
DOE OSTI2021

Electric vehicle fast charging infrastructure planning in urban networks considering daily travel and charging behavior

Electric vehicles are a sustainable substitution to conventional vehicles. This work introduces an integrated framework for urban fast charging infrastructure to address the range anxiety issue. A mesoscopic simulation tool is developed to generate trip trajectories, and simulate charging behavior based on various trip attributes. The resulting charging demand is the key input to a mixed-integer nonlinear program that seeks charging station configuration. The model minimizes the total system cost including charging station and charger installation costs, and charging, queuing, and detouring delays. The problem is solved using a decomposition technique incorporating a commercial solver for small networks, and a heuristic algorithm for large-scale networks, in addition to the Golden Section method. The solution quality and significant superiority in the computational efficiency of the decomposition approach are confirmed in comparison with the implicit enumeration approach. Furthermore, the required infrastructure to support urban trips is explored for future market shares and technologies.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2022

Nth-plant scenario for forest resources and short rotation woody crops: Biorefineries and depots in the contiguous US

Estimating the US potential of woody material is of vital importance to ensure cost-effective supply logistics and develop a sustainable bioenergy and bioproducts industry. We analyzed a mature conversion technology for woody resources for the contiguous US that takes advantage of economies of scale: the nth-plant. Here, we developed a database to quantify the total accessible woody biomass within a distributed network of preprocessing depots and biorefineries considering both quality specifications for conversion and a target cost to compete with fossil fuels. We considered two categories of woody biomass: 1) forest residues from trees, tops and limbs produced from conventional thinning and timber harvesting operations as well as non-timber tree removal; and 2) short rotation woody crops such as poplar, willow, pine, and eucalyptus. A mixed integer linear programming model was developed to analyze scenarios with woody feedstock blends at variable biomass ash contents and cost targets at the biorefinery. When considering a target cost of 85.51 dollars/dry ton (2016$) at the biorefinery, the maximum accessible biomass from forest residues in 2040 remained constant at 106 million dry tons regardless of ash targets. Including short rotation woody crops as part of the blend increased the total accessible biomass to 153 and 195 million dry tons at ash targets of 1% and 1.75%, respectively. We concluded from our analysis that woody resources could address about 55% of EPA’s (Environmental Protection Agency) target of 16 billion gallons of cellulosic biofuel.

09 BIOMASS FUELS↗
DOE OSTI2020

Benders Cut Classification via Support Vector Machines for Solving Two-Stage Stochastic Programs

In this work, we consider Benders decomposition for solving two-stage stochastic programs with complete recourse based on finite samples of the uncertain parameters. We define the Benders cuts binding at the final optimal solution or the ones significantly improving bounds over iterations as valuable cuts. We propose a learning-enhanced Benders decomposition (LearnBD) algorithm, which adds a cut classification step in each iteration to selectively generate cuts that are more likely to be valuable cuts. The LearnBD algorithm includes two phases: (i) sampling cuts and collecting information from training problems and (ii) solving testing problems with a support vector machine (SVM) cut classifier. We run the LearnBD algorithm on instances of capacitated facility location and multicommodity network design under uncertain demand. Our results show that SVM cut classifier works effectively for identifying valuable cuts, and the LearnBD algorithm reduces the total solving time of all instances for different problems with various sizes and complexities.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Stochastic Unit Commitment: Model Reduction via Learning

As weather-dependent renewable generation increases its share in the generation mix of most electric energy systems, a stochastic unit commitment becomes the natural day-ahead scheduling tool. However, such a tool is generally computationally intractable if a detailed uncertainty description is considered. Taking this into account, we proposed a learning method to make the stochastic unit commitment problem tractable. Here, recent advances in statistical learning and machine learning to address optimization problems can be advantageously applied to the rather intractable stochastic unit commitment problem. Considering these advances, we explore simple learning techniques to drastically reduce the size of a stochastic unit commitment problem without significantly altering its optimal solution. The considered stochastic unit commitment problem is formulated as a two-stage stochastic programming problem. The first stage represents commitment decisions, while the second one represents the operation conditions under different scenarios. Taking into account historical solved instances (or proxies for them), we reduce the size (measured by numbers of constraints and variables) of the stochastic unit commitment problem by (i) fixing unchanged binary variables and by (ii) eliminating inactive inequality constraints. Our numerical results show that the reduced problem generally requires significantly less time to solve while obtaining high-quality solutions, which are very close to or indistinguishable from the one obtained by solving the original problem. We use an Illinois 200-bus system to illustrate and characterize the performance of the proposed problem-reduction method.

42 ENGINEERING↗
DOE OSTI2020

Control-Oriented Modeling of Lithium-Ion Batteries

Battery management systems (BMSs), which monitor and optimize performance while ensuring safety, require control-oriented models, i.e., models tailored to the design and implementation of estimation and control algorithms. Physics-based electrochemical models describe detailed battery phenomena, but are too computationally intensive for use in estimation and control applications. Single particle models (SPMs), which retain some of the physics of electrochemical models, are often used for control-oriented battery modeling since they are computationally efficient; however, they are only valid over very low frequency ranges and C-rates. Empirical equivalent circuit models (ECMs) are also used for control-oriented battery modeling since they are computationally efficient and can describe battery behavior over wide frequency ranges; however, they provide no physical understanding of the battery and, therefore, have limited applicability. Further, fractional order terms (e.g., Warburg impedances) are often employed, making the models unwieldy for use in the time domain. This work provides a control-oriented battery model that combines the benefits of SPM and ECM models, while overcoming their limitations. The proposed model incorporates some of the battery physics found in electrochemical models, can easily be used in both the time and frequency domains, and describes battery behavior over its entire frequency range. A linearized single particle model, which incorporates key electrochemical parameters, is used for modeling battery physics at very low frequencies. For low frequencies, integer-order linear systems are used to approximate diffusion physics described by Warburg impedances, and high frequency behavior is modeled by the double layer capacitance effect. The proposed battery model is more computationally efficient than full electrochemical models since it does not require the solution of PDEs, is accurate for a wider frequency range than the SPM considered in this paper, and does not suffer from the unwieldiness and limited applicability of empirical ECMs. Finally, the model is validated in the time and frequency domains via a comparison to pseudo-two-dimensional (P2D) model simulations and experimental data.

25 ENERGY STORAGE↗
DOE OSTI2023

The value of concentrating solar power in ancillary services markets

Ancillary services, such as spinning reserves, can provide grid reliability and contribute to profitability of an energy resource. We exercise an existing dispatch optimization model to estimate the profitability of a concentrating solar power plant by incorporating the sale of spinning reserves in the ancillary service market using the National Renewable Energy Laboratory's System Advisor Model to simulate operations within a 72-h rolling horizon framework. Assuming a price-taker approach with day-ahead energy and spinning reserve prices from both the California Independent System Operator and the Electricity Reliability Council of Texas, we find that selling spinning reserves in addition to electric energy increases plant profitability by up to 7% with perfect knowledge of day-ahead pricing and solar resource availability. Here, this finding suggests that spinning reserve markets provide significant value streams to concentrating solar power plants that can leverage thermal energy storage to offer reliable production in the short-to-medium term.

14 SOLAR ENERGY↗
DOE OSTI2026

A Decomposition-Based Learn-To-Optimize Approach with Feasibility Layer Assistance for Sub-Hourly Unit Commitment

Sub-hourly unit commitment (UC) with 15-min intervals is gaining significant attention as a way to respond rapidly to the fluctuations in electricity supply and demand introduced by renewable resources. However, the increased temporal resolution and complex inter-temporal dependencies pose substantial computational challenges for traditional optimization methods. To this end, this paper explores a decomposition-based learn-to-optimize approach. Building on recent advances in machine learning, our method revisits the long- overlooked Lagrangian relaxation framework, which is a classical decomposition technique that enables tractable subproblem solving. These smaller subproblems are inherently well-suited for machine learning, as their reduced dimensionality and structural regularity allow predictive models to efficiently learn and generalize solution patterns. We thus propose a generic predictive model, which embeds Gated Recurrent Units (GRUs) and Attention in the encoder-decoder structure, and integrate a rule-based feasibility layer to capture temporal dependencies, reduce training effort, and improve feasibility w.r.t. unit-level constraints. Our method has been validated on the IEEE 118-bus system, demonstrating promising performance in solving sub-hourly UC problems efficiently and feasibly.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Nth-plant scenario for blended pellets of Miscanthus, Switchgrass, and Corn Stover using multi-modal transportation: Biorefineries and depots in the contiguous U.S.

The sustainability of the biofuel industry depends on the development of a mature conversion technology on a national level that can take advantage of the economies of scale: the nth-plant. Here, this study addresses the logistic challenge of mobilizing national cellulosic feedstock supplies for a sustainable bioenergy industry. A Mixed Integer Linear Programming (MILP) model was developed and updated to deliver on-spec biomass that considers both a desired quantity and quality at the biorefinery. Our supply chain analysis includes multi-modal transport (truck and rail), varying depot and biorefinery sizes, and feedstock blends of corn stover (harvested by either a two- or three-pass method), switchgrass, and miscanthus. The following US states: Illinois, Kansas, Missouri, North Carolina, Oklahoma, Georgia, and Texas were identified as key locations for producing accessible miscanthus. Based on our most optimistic scenario, using trucks as the only transportation mode in 2040 with a cost target of $\$$79/dt, corn stover, switchgrass, and miscanthus could help meet 48% of the EPA target, 173 million dry tons that translate into 7.8 billion GGE. The addition of rail transportation for biomass delivery to biorefineries could help meet 79% of the EPA target, 283 million dry tons that translate into 12.7 billion GGE.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2022

Building Load Control Using Distributionally Robust Chance-Constrained Programs with Right-Hand Side Uncertainty and the Risk-Adjustable Variants

Aggregation of heating, ventilation, and air conditioning (HVAC) loads can provide reserves to absorb volatile renewable energy, especially solar photo-voltaic (PV) generation. In this paper, we decide HVAC control schedules under uncertain PV generation, using a distributionally robust chance-constrained (DRCC) building load control model under two typical ambiguity sets: the moment-based and Wasserstein ambiguity sets. We derive mixed integer linear programming (MILP) reformulations for DRCC problems under both sets. Especially, for the Wasserstein ambiguity set, we use the right-hand side (RHS) uncertainty to derive a more compact MILP reformulation than the commonly known MILP reformulations with big-M constants. All the results also apply to general individual chance constraints with RHS uncertainty. Furthermore, we propose an adjustable chance-constrained variant to achieve tradeoff between the operational risk and costs. We derive MILP reformulations under the Wasserstein ambiguity set and second-order conic programming (SOCP) reformulations under the moment-based set. Using real-world data, we conduct computational studies to demonstrate the efficiency of the solution approaches and the effectiveness of the solutions. Summary of Contribution: The problem studied in this paper is motivated by a building load control problem that uses the aggregation of heating, ventilation, and air conditioning (HVAC) loads as flexible reserves to absorb uncertain solar photovoltaic (PV) generation. The problem is formulated as distributionally robust chance-constrained (DRCC) programs with right-hand side (RHS) uncertainty. In addition, we propose a risk-adjustable variant of the DRCC programs, where the risk level, instead of being predetermined, is treated as a decision variable. The paper aims to provide tractable reformulations and solution algorithms for both the (general) DRCC and the (general) adjustable DRCC models with RHS uncertainty.

97 MATHEMATICS AND COMPUTING↗