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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

The development of Gibbs's dyadic and implications for the gradient of a vector field

In this paper, we review the history of the dyadic as developed by Gibbs. This mathematical construct appeared in the second part of Gibbs's pamphlet on vector analysis (published in 1884), and it represented the first known development of a Cartesian theory of tensors. Gibbs made extensive use of the dyadic to express his theory of linear vector functions, that is, functions that acted on vectors and mapped them to new vectors. The dyadic proved to be a capable vehicle in Gibbs's hands, and his theory for dyadics (which we would now call second-order Cartesian tensors) was relatively advanced. The theory detailed notions such as the decomposition of vectors and conditions under which a tensor would have an inverse. While Gibbs's theory for linear operators expressed by dyadics was robust, it did not seem to garner the attention that the more conventional vector analysis (published in the first half of his pamphlet in 1881) did. Perhaps in part because of the general unfamiliarity with the dyadic, two distinct and conflicting definitions of the gradient of a vector field have arisen in the literature. The details of these differences in notation, possible reasons for the difference, and a potential resolution are proposed.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Evolution of coupled weakly driven waves in a dissipative plasma

The nonlinear collisional dynamics of coupled driven plasma waves in the presence of background dissipation is studied analytically within kinetic theory. Sufficiently near marginal stability, phase space correlations are poorly preserved and time delays become unimportant. The system is then shown to be governed by two first-order coupled autonomous differential equations of cubic order for the wave amplitudes and two complementary first-order equations for the evolution of their phases. That system of equations can be decoupled and further simplified to a single second-order differential equation of Liénard's type for each amplitude. Numerical solutions for this equation are obtained in the general case, while analytic solutions are obtained for special cases in terms of parameters related to the spacing of the resonances of the two waves in frequency space, e.g., wave lengths and oscillation frequencies. These parameters are further analyzed to find classes of quasi-steady saturation and pulsating scenarios. To classify equilibrium points, local stability analysis is applied, and bifurcation conditions are determined. When the two waves saturate at similar amplitude levels, their combined signal is shown to invariably exhibit amplitude beating and phase jumps of nearly π. In conclusion, the obtained analytical results can be used to benchmark simulations and to interpret eigenmode amplitude measurements in fusion experiments.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Field redefinitions and infinite field anomalous dimensions

Field redefinitions are commonly used to reduce the number of operators in the Lagrangian by removing redundant operators and transforming to a minimal operator basis. We give a general argument that such field redefinitions, while leaving the S-matrix invariant and consequently finite, lead not only to infinite Green’s functions, but also to infinite field anomalous dimensions γ ϕ . These divergences cannot be removed by counterterms without reintroducing redundant operators.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Improving the five-point bootstrap

We present a new algorithm for the numerical evaluation of five-point conformal blocks in d-dimensions, greatly improving the efficiency of their computation. To do this we use an appropriate ansatz for the blocks as a series expansion in radial coordinates, derive a set of recursion relations for the unknown coefficients in the ansatz, and evaluate the series using a Padé approximant to accelerate its convergence. We then study the 〈σσϵσσ〉 correlator in the 3d critical Ising model by truncating the operator product expansion (OPE) and only including operators with conformal dimension below a cutoff ∆ ⩽ ∆cutoff. We approximate the contributions of the operators above the cutoff by the corresponding contributions in a suitable disconnected five-point correlator. Using this approach, we compute a number of OPE coefficients with greater accuracy than previous methods.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Fractional Laplacian spectral approach to turbulence in a dusty plasma monolayer

This work presents an analytical investigation of anomalous diffusion and turbulence in a dusty plasma monolayer, where energy transport across scales leads to the spontaneous formation of spatially disordered patterns. Many-body simulations of 10,000-particle dusty plasma monolayers are used to demonstrate how the global dynamics depend on the statistical properties of the dust assembly for realistic laboratory conditions. We find that disorder due to variations in the dust size distribution and charge-driven nonlocal interactions resulting in anomalous dust diffusion are key factors for the onset of instabilities. The resulting dynamics exhibit features of inertial turbulence over slightly more than half a decade of scales proportional or smaller than the Debye shielding length. These processes are examined analytically using a recently developed Fractional Laplacian Spectral (FLS) technique, which identifies the active energy channels as a function of scale, disorder concentration, and features of the nonlocal-interactions. The predictions from the theoretical (spectral) analysis demonstrate agreement with the results from the many-body (kinetic) simulations, thus providing a powerful tool for the study of active turbulence.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Hilbert series, the Higgs mechanism, and HEFT

We expand Hilbert series technologies in effective field theory for the inclusion of massive particles, enabling, among other things, the enumeration of operator bases for non-linearly realized gauge theories. We find that the Higgs mechanism is manifest at the level of the Hilbert series, as expected for the partition function of an S-matrix that is subject to the Goldstone equivalence theorem. In addition to massive vectors, we detail how other massive, spinning particles can be studied with Hilbert series; in particular, we spell out the ingredients for massive gravity in general spacetime dimensions. Further methodology is introduced to enable Hilbert series to capture the effect of spurion fields acquiring vevs. We apply the techniques to the Higgs Effective Field Theory (HEFT), providing a systematic enumeration of its operator basis. This is achieved both from a direct and a custodial symmetry spurion-based approach; we compare and contrast the two approaches, and our results to those appearing in previous literature.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Evidence of Skewness and Sub-Gaussian Character in Temperature-Dependent Distributions of One Million Electronic Excitation Energies in PbS Quantum Dots

Obtaining statistical distributions by sampling a large number of conformations is vital for an accurate description of temperature-dependent properties of chemical systems. However, constructing distributions with 105??106 samples is computationally challenging because of the prohibitively high computational cost of performing rst-principles quantum-mechanical calculations. In this work, we present a new technique called the Eective Stochastic Potential Conguration Interaction Singles (ESP-CIS) method to obtain excitation energies. The ESP-CIS method uses random matrix theory for the construction of an eective stochastic representation of the Fock operator and combines it with linear-response theory. Here, we perform excitation calculations on PbS quantum dots (0.75-1.75 nm) at temperatures of 200-400 K. Results from a total of 15 million excitation energy calculations (1 million samples for each of the three dots at ve dierent temperatures) revealed the distributions to be sub-Gaussian in nature with negative skewness. The distributions progressively became red-shifted with increasing temperature. The study shows that random distortions in structure from the minimum-energy geometry of these quantum dots are more likely to lower the excitation energies.

Scher, Jeremy A.↗
DOE OSTI2021

Bilinear dynamic mode decomposition for quantum control

Abstract Data-driven methods for establishing quantum optimal control (QOC) using time-dependent control pulses tailored to specific quantum dynamical systems and desired control objectives are critical for many emerging quantum technologies. We develop a data-driven regression procedure, bilinear dynamic mode decomposition (biDMD), that leverages time-series measurements to establish quantum system identification for QOC. The biDMD optimization framework is a physics-informed regression that makes use of the known underlying Hamiltonian structure. Further, the biDMD can be modified to model both fast and slow sampling of control signals, the latter by way of stroboscopic sampling strategies. The biDMD method provides a flexible, interpretable, and adaptive regression framework for real-time, online implementation in quantum systems. Further, the method has strong theoretical connections to Koopman theory, which approximates nonlinear dynamics with linear operators. In comparison with many machine learning paradigms minimal data is needed to construct a biDMD model, and the model is easily updated as new data is collected. We demonstrate the efficacy and performance of the approach on a number of representative quantum systems, showing that it also matches experimental results.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

A perturbative CFT dual for pure NS–NS AdS 3 strings

We construct a conformal field theory dual to string theory on AdS 3 with pure NS–NS flux. It is given by a symmetric orbifold of a linear dilaton theory deformed by a marginal operator from the twist-2 sector. We compute two- and three-point functions on the CFT side to 4th order in conformal perturbation theory at large N. They agree with the string computation at genus 0, thus providing ample evidence for a duality. We also show that the full spectra of both short and long strings on the CFT and the string side match. Furthermore, the duality should be understood as perturbative in N –1 .

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Sensor enabled data-driven predictive analytics for modeling and control with high penetration of DERs in distribution systems

The electric power grid is undergoing a tremendous transformation due to the increasing penetration of renewable energy resources beginning with wind and more recently with the distributed energy resources (DERs) such as solar and battery storage. DERs have dramatically changed the role of the distribution systems in the overall power grid, and they are expected to contribute a significant portion of power generation in the future. If current trends for DERs continue, system operation and control will need to change dramatically for improved grid reliability and resiliency. As renewable resources increase in penetration, new and challenging operational, planning, and design problems are expected to emerge. Some of the key challenges that arise in the planning and operation of the future grid are: 1) Quantifying the impact of high DER penetration in distribution systems on bulk grid behavior over multiple time scales. 2) Identifying whether a particular DER configuration/settings have a large impact on the overall grid behavior. These challenges can be addressed in an offline manner using detailed T&D grid models and they can also be addressed in an online manner using sensor measurements. In particular, the advancement and planned growth in sensor technology in power grid over various voltage levels provide us with a unique opportunity to tackle these challenges from a data analytic perspective without needing detailed T&D grid models. A few questions that naturally arise when addressing the challenges from DERs using sensor data are: 1) How can we use limited sensor measurements to monitor & control voltage stability and small signal stability of the bulk system? 2) How can we ensure that the developed data analytic methods are robust to data availability and quality issues? 3) How can we compute the developed analytics in a scalable manner using streaming measurements? In this project, we addressed the aforementioned challenges arising from DERs and answered the questions raised above on how to effectively use the sensor measurements to enhance the reliability and performance of the electric grid. Thus, the overarching goal of this project is to develop effective reduced/representative system models from data that make the computational complexity sufficiently manageable so as to be useful to simulate, analyze, and even control complex non-linear power systems dynamics with large penetrations of DERs. In order to achieve the objective, the project team established a four-fold technical approach 1) Formulated a combined transmission-distribution co-simulation framework for data generation and validation, 2) Derived reduced/representative models of power systems based on data-driven methods for efficient computation and appropriate representation of system behavior, 3) Developed data driven characterization of power system behavior based on transfer operator theory, machine learning and optimization for model estimation, 4) Incorporated a scalable data management and processing architecture using distributed Kafka streaming applications that coordinate input data streams to the developed data analytics. The key accomplishments of the project are: 1) Development of a scalable multi-timescale T&D co-simulation framework (both for steady state and for dynamic co-simulation) using commercial solvers (PSSE and GridLAB-D). The steady-state T&D co-simulation interface is shared with our industry partner (PJM). 2) A structured reduced order dynamic model of distribution systems that can represent partial motor stalling along with a systematic procedure to derive the model parameters. 3) A PMU based online method to monitor, localize and mitigate fault-induced delayed voltage recovery using DER reactive support and load control in distribution systems. 4) Development of linear operator based robust methodologies for dynamic state estimation, uncertainty quantification, system identification and trajectory prediction for power system dynamics. 5) An adaptive damping control for utilizing wind energy resources to provide oscillation damping and system stability. 6) Implementation of Kafka-based framework for efficient processing of streaming data using Linux-based local virtual environment.

DER integration↗
DOE OSTI2023

Amplitude/operator basis in chiral perturbation theory

We establish a systematic construction of the on-shell amplitude/operator basis for Chiral Perturbation Theory (ChPT) in D = 4 spacetime dimensions and with an arbitrary number of flavors N f . For kinematic factors, we employ spinor-helicity variables to construct the soft blocks, which are local amplitudes satisfying the Adler’s zero condition, as well as to take into account the reduction in the kinematic basis due to the Gram determinant, which arises at O(p 10 ) when the number of multiplicity N in an amplitude becomes large: N > D. For flavor factors, we include group-theoretic relations at small N f , N f ≤ N, which decreases the flavor basis. The result is obtained by adapting the Young tensor method of constructing the operator basis for generic effective field theories to the case of non-linearly realized symmetries. Working in the massless quark limit, we present purely mesonic operators for both even- and odd-parity at O(p 6 ) and O(p 8 ) for N = 6 and arbitrary N f , and establish a direct correspondence between the amplitude basis and the operator basis. Furthermore, the redundancy due to the Gram determinant is studied at O(p 10 ) for N = 6, 8 and 10.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Effective Field Theory for the perturbations of a slowly rotating black hole

We develop the effective theory for perturbations around black holes with scalar hair, in two directions. First, we show that the scalar-Gauss-Bonnet theory, often used as an example exhibiting scalar black hole hair, can be deformed by galileon operators leading to order unity changes to its predictions. The effective theory for perturbations thus provides an efficient framework for describing and constraining broad classes of scalar-tensor theories, of which the addition of galileon operators is an example. Second, we extend the effective theory to perturbations around an axisymmetric, slowly rotating black hole, at linear order in the black hole spin. We also discuss the inclusion of parity-breaking operators in the effective theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2023

Analytical Large-Signal Modeling of Inverter-Based Microgrids With Koopman Operator Theory for Autonomous Control

The microgrid (MG) plays a crucial role in the energy transition, but its nonlinearity presents a significant challenge for large-signal power systems studies in the electromagnetic transient (EMT) time scale. In this paper, we develop a large-signal linear MG model that considers the detailed dynamics of the primary and zero-control levels based on the Koopman operator (KO) theory. Firstly, a set of observable functions is carefully designed to capture the nonlinear dynamics of the MG. The corresponding linear KO is then analytically derived based on these observables, resulting in the linear representation of the original nonlinear MG with observables as the new coordinate. The influence of external input on the system dynamics is also considered during the derivation, enabling control of the MG. We solve the voltage control problem using the traditional linear quadratic integrator (LQI) method to demonstrate that textbook linear control techniques can accurately control the original nonlinear MG via the developed KO-linearized MG model. Our proposed KO linearization method is generic and can be easily extended for different control objectives and MG structures using our analytical derivation procedure. Here, we validate the effectiveness of our methodology through various case studies.

25 ENERGY STORAGE↗
DOE OSTI2025

Optimal Network Reconfiguration and Scheduling With Hardware-in-the-Loop Validation for Improved Microgrid Resilience

With the increased occurrence of various major extreme weather events, power outages and prompt power system restorations have recently drawn more attention to the resilience and recovery of power systems. From the perspective of a more resilient power delivery at the distribution grid, system restoration using network topology reconfiguration together with optimal scheduling of distributed energy resources are adopted in this paper. The proposed optimization model aims at minimizing the total load shedding cost and other operational costs, in which linearized topological constraints borrowed from graph theory and linearized DistFlow models are respectively used to maintain the radial network topology and power flow balance after system contingencies. To demonstrate the applicability of the proposed strategy, a real-world case study of a networked three-microgrid system in Adjuntas, Puerto Rico, is used with the consideration of different independent/interconnected microgrid scenarios, contingencies, and fairness settings. Furthermore, hardware-in-the-loop testing is conducted for the same three-microgrid network, where the closely matched results with the simulated ones have validated the effectiveness of the proposed restoration strategy, which is now ready to move one step forward towards field deployment. Finally, to test the proposed restoration strategy in a larger networked system, the modified IEEE-33 bus test distribution system is considered, and the results show a more resilient power delivery for critical loads under three and four line outages.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2020

A comprehensive conductivity model for drift and micro-tearing modes

The parallel electrical conductivity is a crucial parameter in the study of the linear stability of drift-modes like the micro-tearing mode (MTM). The conductivity enters by closing the electromagnetic tearing layer equations. Recent progress in the understanding of the pedestal suggests that the MTM could play an important role in its structure and evolution. For this reason, we revisit and improve previous model conductivities. This parameter is generally derived from the linearized drift kinetic equation. In the past literature, it has been computed using either simplified collision operators or neglecting the spatial dependence away from the rational surface. Here, a fully consistent expression for the conductivity that would accurately model the pedestal has not been available. By applying a novel variational procedure and with the full Fokker Plank collision operator, including electron–electron collisions, we compute a closed expression for the parallel electrical conductivity in the form of a rational function.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Spin-free formulation of the multireference driven similarity renormalization group: A benchmark study of first-row diatomic molecules and spin-crossover energetics

We report a spin-free formulation of the multireference (MR) driven similarity renormalization group (DSRG) based on the ensemble normal ordering of Mukherjee and Kutzelnigg (1997). This ensemble averages over all microstates of a given total spin quantum number, and therefore, it is invariant with respect to SU(2) transformations. As such, all equations may be reformulated in terms of spin-free quantities and they closely resemble those of spin-adapted closed-shell coupled cluster (CC) theory. The current implementation is used to assess the accuracy of various truncated MR-DSRG methods (perturbation theory up to third order and iterative methods with single and double excitations) in computing the constants of 33 first-row diatomic molecules. The accuracy trends for these first-row diatomics are consistent with our previous benchmark on a small subset of closed-shell diatomic molecules. We then present the first MR-DSRG application on transition-metal complexes by computing the spin splittings of the [Fe(H 2 O) 6 ] 2+ and [Fe(NH 3 ) 6 ] 2+ molecules. A focal point analysis (FPA) shows that third-order perturbative corrections are essential to achieve reasonably converged energetics. The FPA based on the linearized MR-DSRG theory with one- and two-body operators and up to a quintuple- ζ basis set predicts the spin splittings of [Fe(H 2 O) 6 ] 2+ and [Fe(NH 3 ) 6 ] 2+ to be –35.7 and –17.1 kcal mol –1 , respectively, showing good agreement with the results of local CC theory with singles, doubles, and perturbative triples.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Propagating Parameter Uncertainty in Power System Nonlinear Dynamic Simulations Using a Koopman Operator-Based Surrogate Model

In this work, we propose a Koopman operator-based surrogate model for propagating parameter uncertainties in power system nonlinear dynamic simulations. First, we augment a priori known state-space model by reformulating parameters deemed uncertain as pseudo-state variables. Then, we apply the Koopman operator theory to the resulting state-space model and obtain a linear dynamical system model. This transformation allows us to analyze the evolution of the system dynamics through its Koopman eigenfunctions, eigenvalues, and modes. Of particular importance for this letter, the obtained linear dynamical system is a surrogate that enables the evaluation of parameter uncertainties by simply perturbing the initial conditions of the Koopman eigenfunctions associated with the pseudo-state variables. Simulations carried out on the New England test system reveal the excellent performance of the proposed method in terms of accuracy and computational efficiency.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2020

Application of Koopman operator for model-based control of fracture propagation and proppant transport in hydraulic fracturing operation

This work explores the application of the recently developed Koopman operator approach for model identification and feedback control of a hydraulic fracturing process. Controlling fracture propagation and proppant transport with precision is a challenge due in large part to the difficulty of constructing approximate models that accurately capture the characteristic moving boundary and highly-coupled dynamics exhibited by the process. Koopman operator theory is particularly attractive here as it offers a way to explicitly construct linear representations for even highly nonlinear dynamics. The method is data-driven and relies on lifting the states to an infinite-dimensional space of functions called observables where the dynamics are governed by a linear Koopman operator. Here this work considers two problems: (a) fracture geometry control, and (b) proppant concentration control. In both cases, an approximate linear model of the corresponding dynamics is constructed and used to design a model predictive controller (MPC). The manuscript shows that in the case of highly nonlinear dynamics, as observed in the proppant concentration, use of canonical functions in the observable basis fails. In such cases, a priori system knowledge can be leveraged to choose the required basis. The numerical experiments demonstrate that the Koopman linear model shows excellent agreement with the real system and successfully achieves the desired target values maximizing the oil and gas productivity. Additionally, due to its linear structure, the Koopman models allow convex MPC formulations that avoid any issues associated with nonlinear optimization.

42 ENGINEERING↗