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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 289 records · Page 16
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Electrocatalytic nitrate reduction: controlling adsorbate affinity to tailor reaction products

Every year, Haber-Bosch nitrogen fixation to form ammonia releases immense volumes of CO2. At the same time, nitrate contamination from untreated wastewater threatens human health across the US. Development of circular processes to efficiently upgrade waste nitrate for reuse is critical to sustainably address this growing environmental hazard. Electrochemical reduction of nitrate operates at ambient temperatures and pressures, and can leverage distributed renewable energy sources and water as a hydrogen source for ammonia production. However, current catalysts lack electron efficiency in reducing nitrate versus water, lack selectivity in forming ammonia as a product, or rely on expensive rare metals, making widespread implementation unfeasible. To overcome these limitations, this proposal seeks a mechanistic understanding of nitrate electrochemical reduction on earth abundant metals and their alloys, with the goal of identifying active-site properties that improve both Faradaic efficiency and product selectivity. This insight will extend to other electrocatalytic reductive process that compete with water reduction.

30 DIRECT ENERGY CONVERSION↗
DOE OSTI2025

Evaluating Air-Source Heat Pumps with Distributed Energy Resources in REopt

This training introduces attendees to NREL's REopt(R) web tool and its capabilities in evaluating air-source heat pumps (ASHP) alongside distributed energy resources (DERs). Designed for energy managers, facility operators, and sustainability managers, this session will explore how REopt provides techno-economic analysis to assess the financial viability of ASHP while identifying synergies with solar PV and battery storage. Participants will gain practical insights into using REopt to optimize energy strategies, reduce costs, and improve site resilience.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2025

Networked Microgrid Energy Management Considering Ownership and Control Structures: A Comparison

With the growing deployment of microgrids, networked microgrids have emerged for their additional advantages of economy, reliability and resilience by coordinating the operation of multiple microgrids. As microgrids are operated with different ownership, objectives and functionalities, the formed networked microgrids show characteristics of mixed ownership, inconsistent objectives and various functionalities. To enable the coordinated operation of networked microgrids, three control structures, i.e., centralized, distributed and decentralized, have been constructed in the literature. However, the data sharing enabling these different paradigms and the resulted value propositions are not well defined, leading to poor resource management and resilience, etc. To solve this issue, a complete comparison of networked microgrid energy management under centralized, distributed and decentralized structures are performed. As a novel contribution, the required minimum data exchange of networked microgrid energy management under three control structures are identified, respectively. The value propositions are calculated and compared against each other by the results of case studies.

control structures↗
DOE OSTI2026

Topology-Dependent Performance of Free-Space Photonic Quantum Networks Under Noise

Photonic quantum communication enables secure and high-fidelity information transfer beyond classical limits, with direct relevance to emerging quantum networks operating in free-space environments. While physical-layer models of depolarizing noise, Gamma–Gamma turbulence statistics, entanglement swapping, and decoy-state QKD security bounds are individually well established, prior work typically treats these components in isolation or under fixed network assumptions. In this work, we develop a unified topology-aware analytical framework that simultaneously integrates free-space optical link budgets, turbulence-induced visibility degradation, depolarizing qubit noise, multi-hop entanglement cascade dynamics, teleportation fidelity thresholds, CHSH nonlocality certification, and asymptotic decoy-state secret key rate bounds across star, mesh, and ring graph structures. Rather than introducing new physical channel models, we demonstrate that identical physical links exhibit fundamentally different end-to-end performance once embedded within different network topologies. Mesh architectures minimize visibility cascade through hop-count reduction but incur quadratic hardware scaling. Star topologies minimize link count but concentrate noise and synchronization overhead at the hub. Ring configurations offer linear hardware scaling with multiplicative fidelity degradation. The results establish topology as a first-order design parameter in near-term free-space quantum networks operating without full quantum repeater infrastructures. While motivated by distributed multi-agent architectures, the framework applies broadly to terrestrial, airborne, and satellite-assisted photonic quantum communication systems.

QKD↗
DOE OSTI2025

Economic Evaluation of Modernization Expenditures for Electric Utility Distribution Systems: A Guide for Utility Regulators

Cost-effectiveness evaluation of potential grid modernization solutions is integral to integrated distribution system planning (IDSP). This report synthesizes and updates prior cost-effectiveness approaches identified by the U.S. Department of Energy for grid modernization investments, incorporating multi-objective decision-making. The structure of the report follows the IDSP process, from identifying objectives and priorities to prioritizing grid solutions. The report details two methods for conducting initial cost-effectiveness screening of individual and interdependent grid components: (1) Lowest Reasonable Cost and (2) Benefit-Cost Analysis. Drawing on solutions that passed the initial cost-effectiveness screen, the utility prioritizes grid solutions based on policy objectives and priorities, regulatory compliance, operational efficiencies, and other factors to develop a portfolio of distribution solutions. The utility uses multi-objective decision-making to assess each proposed expenditure against each objective and respective metric. The utility applies a weighting factor, reflecting the priority ranking of the objective, to the total numerical “score” based on the contribution of the proposed solution to addressing each objective. Ultimately, the utility ranks each expenditure from highest to lowest priority using the final score for each solution as well as its cost. The report includes examples of emerging best practices for states and utilities and a cost-effectiveness process checklist.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

Understanding Line Losses and Transformer Losses in Rural Isolated Distribution Systems

Rural, isolated power systems in the mainland U.S. and in states like Alaska and Hawaii are powered by assets like diesel generators. These rural, isolated power systems also cannot operate at the higher band of medium voltage (like 69kV). They are primarily in the 12 to 14 kV range to keep the cost of the distribution investments lower. Because of this mid-band medium voltage range, the line losses and distribution transformers losses consume significant diesel consumption (almost 10 percent of the peak load). This work considers one such power system powering an isolated system and presents key findings online losses, and transformer losses. Understanding and documenting the impacts is critical for these communities operating their power systems and take actions to reduce expensive diesel consumption. In this paper, we will present one such typical grid and model it in electromagnetic transients (EMT) domain. We used the tower structure, under ground cabling installation to develop high fidelity models of lines. We also used high fidelity models of distribution transformers to present the no-load losses and full load loses. We will also present technical solutions available commercially off-the-shelf to reduce these losses and reduce diesel consumption. This work will be a primer for communities to understand the technical challenges and to understand the possible solution available to solve such challenges for rural, isolated power system operators.

blackstart↗
DOE OSTI2026

Continual Learning for Particle Accelerators

Particle accelerators operate under dynamically changing conditions, which often lead to data distribution drifts. These drifts pose significant challenges for Machine Learning (ML) models, which typically fail to maintain performance when faced with such non-stationary data. In particle accelerators, the primary sources of these data drifts include changes in accelerator settings and non-measured parameters such as machine degradation and environmental factors. Previous research has proposed conditional models to handle multiple beam configurations effectively; however, it is challenging to train the ML models on all possible configuration settings. Additionally, conditional models alone can not address performance degradation caused by drifts due to non-measured factors. These limitations contribute to a significant gap between ML development and its deployment in real-world operational settings. To bridge this gap, in this paper, we identify some of the key areas within particle accelerators where continual learning can help mitigate drift-induced performance degradation. In addition, we present a practical use case where a conditional Auto-Encoder model coupled with memory-based continual learning has been employed to demonstrate stable performance even when underlying data drifts.

Schram, Malachi [Thomas Jefferson National Acceler↗
DOE OSTI2026

Continual Learning for Particle Accelerators

Particle accelerators operate under dynamically changing conditions, which often lead to data distribution drifts. These drifts pose significant challenges for Machine Learning (ML) models, which typically fail to maintain performance when faced with such non-stationary data. In particle accelerators, the primary sources of these data drifts include changes in accelerator settings and non-measured parameters such as machine degradation and environmental factors. Previous research has proposed conditional models to handle multiple beam configurations effectively; however, it is challenging to train the ML models on all possible configuration settings. Additionally, conditional models alone can not address performance degradation caused by drifts due to non-measured factors. These limitations contribute to a significant gap between ML development and its deployment in real-world operational settings. To bridge this gap, in this paper, we identify some of the key areas within particle accelerators where continual learning can help mitigate drift-induced performance degradation. In addition, we present a practical use case where a conditional Auto-Encoder model coupled with memory-based continual learning has been employed to demonstrate stable performance even when underlying data drifts

Rajput, Kishansingh [Thomas Jefferson National Acc↗
DOE OSTI2024

Characterizing and improving the performance of molten-salt-steam heat exchangers in concentrating solar power plants

Shell-and-tube heat exchangers (HXs) for steam generation from molten salts in concentrating solar power (CSP) plants experience thermal fatigue due to significant temperature gradients and inherent transient operation. Molten salt-steam HX design lifespans exceed actual lifespans, and, as a consequence, designers overpredict plant profitability and operators neglect appropriate prescriptions to optimize these lifetimes. Here, this study refines HX lifespan estimates with data benchmarked against thermal-fluid mechanical modeling of stress and accumulated fatigue. Reduced-order thermal models of the molten salt-steam, shell-and-tube evaporator and superheater predict transient temperature profiles along the two HXs salt-steam flow paths. The modeled evaporator and superheater temperature profiles enable assessment of cyclic stresses within the HX tubesheets, where molten-salt HX failures are most common. Evaporator and superheater performance data from a current 110 MW elec commercial CSP plant provide a basis for validating the reduced-order HX models. HX life predictions derived from stochastic failure distributions serve as inputs for simulating and optimizing existing plant operations. The impact of the updated lifespans on overall plant revenue depends on operating scenarios. This study suggests that typical ramping rates for a CSP plant with a high-temperature Rankine cycle result in an evaporator and superheater life of approximately 10 and 25 years, respectively, compared to the design target of 30 years. Reduced HX lifespans decrease operational plant revenue on average by 4.6-5.1%. Furthermore, there may be as many as four HX replacements over the 30-year lifetime of the plant; and, purchase agreement loss due to failure to meet contractual production requirements can have ramifications that include the risk of bankruptcy.

14 SOLAR ENERGY↗
DOE OSTI2026

Storage Field Development Plan: One Earth Energy

This Storage Field Development plan presents the Storage Complex characterization results, construction, monitoring, and operational plans, and costs associated with the proposed One Earth Sequestration Carbon Capture and Storage (OES-CCS) site in McLean County, Illinois, near Gibson City. The proposed storage complex, known as the Mt. Simon Storage Complex, comprises the Cambrian Mt. Simon Sandstone reservoir and the primary seal, the Cambrian Eau Claire Formation. The lowermost Underground Source of Drinking Water (USDW) identified for the site is the Ordovician St. Peter Sandstone. Geologic characterization of the Mt. Simon Storage Complex at the OES-CCS site was performed by the Illinois Storage Corridor CarbonSAFE Phase III project, which also prepared and submitted three UIC Class VI applications to construct three injection wells; the permit applications were submitted and are in the federal EPA review process. A characterization well, OEE #1, was drilled to collect site-specific data. These data were analyzed and used to develop the UIC Class VI applications. The OEE #1 well will be converted to an in-zone monitoring (IZM) well for the injection phase. The proposed buildout for the OES-CCS site includes (1) three injection wells (OES #1, OES #2, and OES #3), (2) two IZM wells, (3) two above confining zone (ACZ) monitoring wells, one of which will be used to monitor the lowermost USDW, (4) capture and compression facilities, and (5) transportation facilities, i.e., pipelines. A pre-operational testing program was proposed in the Class VI permit application and will be employed at the site pending approval. Additional pre-injection (baseline), syn-injection, and post-injection monitoring and site care procedures will be followed by OES to ensure that injection activities are protective of human health and the environment. Injection is scheduled to begin in 2025, distributed across the three injection wells in accordance with the permit operating conditions. One Earth Sequestration intends to inject up to 90 million tonnes of CO 2 over a period of approximately 20 years. Injection will begin at approximately 0.5 million tonnes of CO 2 annually and ramp up to a maximum of 4.5 million tonnes annually. Daily injection rates are expected to range from 1,400 to 1,500 tonnes per day initially and reach a maximum of approximately 4,225 tonnes per day, depending on site geology and injectivity at each injection well location, and CO 2 availability. The costs associated with the OES-CCS project include pre-operational costs (e. g. additional seismic data acquisition and well drilling), capture and transportation facility and equipment costs, predicted field operating expenditures (OpEx), and decommissioning and post-injection site care (PISC) costs. The risks associated with project activities, such as site construction, injection operations, and verification of secure storage were evaluated, and mitigation strategies proposed to alleviate those risks.

09 BIOMASS FUELS↗
DOE OSTI2024

Renormalization of asymmetric staple-shaped Wilson-line operators in lattice and continuum perturbation theory

In this work, we study the renormalization of nonlocal quark bilinear operators containing an asymmetric staple-shaped Wilson line at the one-loop level in both lattice and continuum perturbation theory. These operators enter the first-principle calculation of transverse momentum-dependent parton distribution functions (TMDPDFs) in lattice QCD using the formulation of large momentum effective theory. We provide appropriate RI ′ -type conditions that address the power and logarithmic divergences, as well as the mixing among staple operators of different Dirac structures, using a number of different possible projectors. A variant of RI ′ , including calculations of rectangular Wilson loops, which cancel the pinch-pole singularities of the staple operators at infinite length and reduce residual power divergences, is also employed. We calculate at one-loop order the conversion matrix, which relates the quasi-TMDPDFs in the RI ′ -type schemes to the reference scheme MS ¯ for arbitrary values of the renormalization momentum scale and of the dimensions of the staple. Published by the American Physical Society 2024

Astronomy & Astrophysics↗
DOE OSTI2025

Resilient Operation of Networked Community Microgrids with High Solar Penetration

This project, funded by the US Department of Energy’s Solar Energy Technologies Office (SETO), focused on the operation of microgrids as a coordinated network. The primary objective, which was successfully achieved, was to develop both control strategies and hardware solutions to support the resilient and efficient operation of networked microgrids with high solar penetration. The work was structured around the following four main tasks: • Development of distributed and scalable optimization algorithms for AC-coupled networked microgrids. • Design and implementation of a novel DC interconnection hardware to enable precise power exchange between microgrids. • Laboratory operational validation of the developed technologies using 480 V testbeds and commercially available hardware. • Field operational validation of the complete solution in Adjuntas, Puerto Rico, interconnecting two kW-scale, split-phase microgrids of Casa Pueblo’s microgrids. This project addressed multiple technical challenges across the domains of optimization, control, hardware interconnection, and protection. One of its key contributions was delivering tangible, real-world solutions for networking microgrids. In contrast to purely theoretical or simulation-based work, this project included full-scale hardware operational validation both in the lab and in the field. The work conducted as part of this project—in collaboration with the University of Puerto Rico; the University of Tennessee, Knoxville; the University of Central Florida; and Casa Pueblo—has advanced the state of the art in networked microgrids. Key contributions include the development of distributed control strategies, practical solutions for real-world implementation challenges, and the introduction of a novel DC interlink approach for microgrid interconnection. The project featured both laboratory and field validation using commercial off-the-shelf components. The field deployment successfully validated that a group of microgrids can operate in a coordinated manner, enabling precise power flow between systems and mutual support during extreme events. This project resulted in 15 journal publications and 15 conference papers; 5 graduate students and 15 undergraduate students were supported. The codes of distributed optimization and forecasting were made open-source through OSTI.gov for distributed optimization and forecasting. All the publications are available in the ORNL-hosted project landing page. The DC interlink with state-of-charge balancing control was operationally validated in Adjuntas by interconnecting two real-world, 240 V split-phase microgrids. To the best knowledge of the team, this represents the first operational validation of AC microgrids interconnected via DC-interlinks. As a culmination of this project, a follow-on grant was awarded to support the technology transfer of the distributed optimization framework to a commercial microgrid controller, Stellar Edge, developed by the California-based company New Sun Road.

14 SOLAR ENERGY↗
DOE OSTI2025

PowderJet: Spherical metal powder production via multi-orifice droplet-on-demand metal jetting

Leading metal additive manufacturing techniques, such as laser powder bed fusion and directed energy deposition, rely on high-quality spherical metal powders. However, traditional powder production methods like gas atomization face limitations, including low in-spec yield, asphericity, and internal porosity. We introduce PowderJet, a powder production platform that uses electromagnetic pulses to eject liquid metal droplets from a multi-orifice nozzle. Unlike stochastic methods, PowderJet tightly controls powder size, distribution, and purity through a droplet-on-demand approach. We detail the system’s design, operation, and performance using a combined experimental and computational fluid dynamics (CFD) framework. Initial results with Al4008 and Cu110 alloys demonstrate successful production, yielding unsieved aluminum powder batches with a mean diameter of 200 µm and a narrow size distribution (15 µm standard deviation). The produced powders are highly spherical, achieving a roundness > 0.95. PowderJet operates with a small melt volume (3 mL) and supports continuous refilling, enabling production rates between 30 and 140 cm³/hr depending on jetting frequency, number of orifices and particle size. CFD simulations show that future systems could achieve rates exceeding 1000 cm³/hr for particle sizes as small as 40 µm. PowderJet’s high yield of in-spec powder makes it ideal for producing precious or hazardous materials that are inefficient to manufacture using conventional methods. This platform offers a scalable, precise, and efficient solution for producing high-quality powders tailored for advanced manufacturing applications.

Atomization↗
DOE OSTI2026

Online task-space motion control for positioner-coordinated multi-robot manufacturing systems

Incorporating multiple robotic manipulators into large-scale manufacturing systems enhances production efficiency and expands manufacturing capabilities beyond those of single-robot systems. Workpiece positioners in robotic manufacturing have demonstrated significant benefits for process optimization, but coordination strategies for multi-robot systems with shared positioners have received limited attention. This work presents a task-space coordinated trajectory-tracking control framework for multi-robot manufacturing systems, in which robots coordinate their motions within a shared, dynamic workpiece positioning frame. A workpiece positioner actively adjusts the pose of the manufactured component to enable greater operational concurrency and improve overall production efficiency. The proposed motion-coordination scheme employs a distributed and scalable architecture, supporting coordination across heterogeneous multi-robot systems. Two optimization methodologies are introduced to manage kinematic redundancies and maintain continuous, near-optimal operation throughout the manufacturing process. The first strategy exploits a task-space dimensionality reduction to achieve locally optimal configurations by leveraging symmetry-axis rotations of the tool. The second strategy utilizes the workpiece positioner to drive the coordinated robots toward stable and kinematically favorable configurations. For both optimization strategies, multiple objectives are defined to improve key performance metrics, including manipulability, configuration consistency, proximity to mechanical limits, and motion efficiency. Addressing a key limitation of existing coordination approaches, the framework is designed around online setpoint modification, allowing coordinated robots to respond effectively to in-situ process feedback. The proposed control framework is validated using the Robot Operating System (ROS) middleware on a combination of physical and simulated multi-robot system hardware.

Arbogast, Alex [ORNL] (ORCID:0000000154740723)↗
DOE OSTI2026

Omega-Limit Sets and Input-To-State Stability in Power Grids with Switching Equilibria

This paper studies a power transmission system with both conventional generators (CGs) and distributed energy assets (DEAs) providing frequency control. We consider an operating condition with demand aggregating two dynamic components: one that switches between different values on a finite set, and one that varies smoothly over time. Such dynamic operating conditions may result from protection scheme activations, external cyber-attacks, or due to the integration of dynamic loads, such as data centers. Mathematically, the dynamics of the resulting system are captured by a system that switches between a finite number of vector fields - or modes -, with each mode having a distinct equilibrium point induced by the demand aggregation. To analyze the stability properties of the resulting switching system, we leverage tools from hybrid dynamic inclusions and the concept of ..omega.. -limit sets from sets. Specifically, we characterize a compact set that is semi-globally practically asymptotically stable under the assumption that the switching frequency and load variation rate are sufficiently slow. For arbitrarily fast variations of the load, we use a level-set argument with multiple Lyapunov functions to establish input-to-state stability of a larger set and with respect to the rate of change of the loads. The theoretical results are illustrated via numerical simulations on the IEEE 39-bus test system.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

Test and Validate Distributed Coaxial Cable Sensors for in situ Condition Monitoring of Coal-Fired Boiler Tubes

This project aims to test, validate, and advance the technology readiness level (from TRL5 to TRL7) of a novel low-cost distributed stainless-steel/ceramic coaxial cable sensing (SSC-CCS) technology for in situ monitoring of the boiler tube temperature in existing coal-fired power plants. The novel SSC-CCS sensing technology and associated condition-based monitoring (CBM) software to be demonstrated in this project will lead to an improved understanding of the boiler tube failure mechanisms and a prognostic system to improve the overall performance, reliability, and flexibility of the nation’s coal-fired power plant fleet. A boiler tube monitoring system with distributed coaxial cable temperature sensors and a sensor acquisition system was constructed. The high-temperature coaxial cable sensor with a length of 1.3m was made by using a quartz tube (1mm inner diameter (ID) and 6mm outer diameter (OD)) to concentrically separate a 304 stainless-steel (SS) rod (1mm OD) and SS tube (7.94mm OD and 6.16mm ID). The sensor acquisition system includes a vector network analyzer (VNA), a radio frequency (RF) power amplifier, multiple switches and a USB hub. The distributed stainless-steel quartz coaxial cable sensor (SSQ-CCS) had a linear response to temperature with a resolution uncertainty of σ = 0.77℃. To withstand the harsh conditions of 3,300 steam pressures and 800℃ high temperatures, the sensor was shielded by a protective tube made of the same material as the boiler tube. The protection tube had an OD of 1.5 inches and a thickness of 0.25 inches. In the laboratory tests, the sensor showed good sensitivity and fast response. The drift was bounded between +0.33% and -0.67% during a test at 600℃ for 350 hours, indicating good stability of the sensor. A field test was conducted where four sensors were welded on four superheat tubes (SH-Ts) at a coal-fired power station over 400 days. Conventional thermocouples were welded to the superheater tubes alongside the coaxial cable sensors for the purpose of comparison. Two sensors were capable of distributed sensing, with three multiplexed sensing sections. The other two sensors were single section. During the 400-day test period, the power plant experienced startups and shutdowns. At the steady state operations, the temperature of the boiler tube is about 600℃ (1112°F). The sensors recorded the entire coal-firing processes (start-up, steady state, and shut-down) and the glitch event. A GSM modem and a Watchdog were added to the system to ensure reliable data recording. The GSM modem sent daily messages to plant managers and Clemson team to inform the status of the sensor system. If the system was not normally working, the Watchdog would reboot the system automatically. The new coaxial cable based distributed sensing technology has been proven to be successful in both laboratory and field tests. A comprehensive four-stage multi-physics computational framework has been developed to assist the design, optimization, installation, and operation of SSQ-CCS. With the consideration of various operation conditions, we predict the distributions of flue gas temperatures within coal-fired boilers, the temperature correlation between the boiler tube and SSQ-CCS, and the safety of SSQ-CCS. A conditional-based monitoring system is implemented as well. The computational framework developed in this work can guide the future operation of coal-fired plants and other power plants for the safety prediction of boiler operations.

01 COAL, LIGNITE, AND PEAT↗
DOE OSTI2025

Dynamic Boundary Microgrids Under Privatization Considerations

Microgrids have physical, electrical, and logical (data, network, and ownership) boundaries. To power unserved customer loads during an outage, microgrids can extend the traditional operational boundaries. This can become complex when considering microgrid-to-microgrid (M2M) interactions where sensitive information such as competitive microgrid operational data is not shared. This work proposes an optimization method coordinated between microgrid controllers and distribution management systems that limits data sharing. The method involves a competitive bidding strategy that maximizes unserved load coverage while minimizing resource utilization and sensitive operational data sharing among entities. The work is validated on a two-microgrid system with photovoltaic and energy storage systems and curves of load derived from real world residential buildings datasets. Results show that the proposed method, when applied for three distinct use cases of energy storage sufficiency to cover the predefined boundary and/or the expanded boundary, can successfully select and bid the available load coverage.

Starke, Michael [ORNL] (ORCID:0000000221211195)↗
DOE OSTI2026

Back-to-back dijet production in DIS at arbitrary Bjorken x: TMD gluon distributions to twist-3 accuracy

We derive the gluon transverse-momentum-dependent (TMD) operator structure of back-to-back\\\\r\\\\nquark–antiquark dijet production in deep inelastic scattering at arbitrary Bjorken-x to twist-3 ac\\\\r\\\\ncuracy. Working at leading order in the strong coupling and in the kinematic regime where the\\\\r\\\\ntransverse momentum imbalance of the jets is much smaller than their individual transverse mo\\\\r\\\\nmenta, we perform a systematic gradient expansion of the quark propagator in a background gluon\\\\r\\\\nfield. This expansion organizes multiple interactions with the target in terms of longitudinal Wilson\\\\r\\\\nlines and gauge-invariant field-strength insertions, yielding a TMD description valid beyond the\\\\r\\\\nstrict high-energy eikonal (x → 0) approximation. We obtain explicit cross sections for longitudi\\\\r\\\\nnally and transversely polarized virtual photons, identifying all contributing gluon TMD operators\\\\r\\\\nup to twist-3, including structures involving F+−, Fij, and three-gluon correlators. The full lon\\\\r\\\\ngitudinal phase eixP+z− associated with Bjorken-x is retained throughout. In the small-x limit,\\\\r\\\\nour results reproduce the known sub-eikonal expressions obtained in the Color Glass Condensate\\\\r\\\\nframework, establishing a direct connection between the general-x TMD expansion and high-energy\\\\r\\\\nfactorization. We further reduce the operator basis using equations of motion, minimizing the num\\\\r\\\\nber of independent nonperturbative matrix elements entering the cross section. This work provides\\\\r\\\\na systematic foundation for extending TMD analyses of dijet production beyond leading twist, es\\\\r\\\\ntablishing a unified operator framework valid at arbitrary Bjorken-x that smoothly interpolates\\\\r\\\\nbetween moderate- and small-x descriptions of gluon TMDs.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗