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At least 1,333 records · Page 74

Simple Forest Canopy Thermal Exitance Model

We describe a model to calculate brightness temperature and surface energy balance for a forest canopy system. The model is an extension of an earlier vegetation only model by inclusion of a simple soil layer. The root mean square error in brightness temperature for a dense forest canopy was 2.5 C. Surface energy balance predictions were also in good agreement. The corresponding root mean square errors for net radiation, latent, and sensible heat were 38.9, 30.7, and 41.4 W/sq m respectively.

Smith J. A.↗

The Thomas theorem and the Efimov States within a generalized Bohr model

Using a generalized Bohr model and the hyper-spherical formalism for a three-body system, we derive the universal energy functions. We apply our model to 4He atom where only the Coulomb potential is dominant and find that its binding energy is well reproduced (less than 5.5% difference). Later, we focus on the equal mass three-body systems and derive the Thomas theorem assuming a simple interaction depending on the range of the potential. We discuss the conditions for which an unbound two-body system produces a bound three-body system and apply our model to 4He and triton atoms as well as to the triton nucleus. Using their scattering lengths and effective ranges, we are able to reproduce the two-body or the three-body binding energies (less than 5% difference) with only one parameter fitted. Prediction for excited (Efimov) levels are also given and in particular we demonstrate that for some hyper-angles two equal minima appear which indicate a phase (shape) transition similar to the Landau's theory of phase transition. We suggest that the observed excited levels in two different experiments for the triton nucleus are indeed Efimov levels and there may be more surprises.

74 ATOMIC AND MOLECULAR PHYSICS↗

A reduced order sulfuric acid decomposition model for a nuclear-powered hybrid sulfur cycle

A reduced order model of sulfuric acid decomposition within a bayonet chemical reactor was developed to support the U. S. Department of Energy Integrated Energy System program, and address the lack of knowledge in scaling and integration for joint chemical and nuclear processes. Sulfuric acid decomposition within a bayonet reactor was modeled to provide chemical and thermodynamic data relevant to advanced nuclear reactor-driven integrated energy systems based on desired operational scale and operational conditions. The temperature range required for high-temperature advanced nuclear reactor integrated energy systems, 750-850 °C, was shown to produce reasonable agreement (within a few percent relative error) with past models and experimental data, and yielded good efficiency results for bayonet reactor operations. The results of the reduced order model agreed with previous work from Savannah River National Labs within a maximum of 3.4% error on the decomposition of sulfur trioxide, and on previous Hybrid Sulfur flowsheets from Gorensek and Summers that showed operational temperature, pressure, and composition ranges for efficiency which made the Hybrid Sulfur cycle competitive with water electrolysis. Here, the agreement with previous high-fidelity models provided a framework for future Integrated Energy System grid evaluations with an advanced nuclear reactor and large-scale hydrogen production using a mathematical model to represent chemical operations.

08 HYDROGEN↗

CERES ERBE-like Monthly Geographical Averages (ES-4) in HDF (CER_ES4_FM1+FM3_Edition2)

The ERBE-like Monthly Geographical Averages (ES-4) product contains a month of space and time averaged Clouds and the Earth's Radiant Energy System (CERES) data for a single scanner instrument. The ES-4 is also produced for combinations of scanner instruments. For each observed 2.5-degree spatial region, the daily average, the hourly average over the month, and the overall monthly average of shortwave and longwave fluxes at the Top-of-the-Atmosphere (TOA) from the CERES ES-9 product are spatially nested up from 2.5-degree regions to 5- and 10-degree regions, to 2.5-, 5-, and 10-degree zonal averages, and to global monthly averages. For each nested area, the albedo and net flux are given. For each region, the daily average flux is estimated from an algorithm that uses the available hourly data, scene identification data, and diurnal models. This algorithm is 'like' the algorithm used for the Earth Radiation Budget Experiment (ERBE). The following CERES ES4 data sets are currently available: CER_ES4_FM1+FM2_Edition1 CER_ES4_PFM+FM1+FM2_Edition1 CER_ES4_PFM+FM1+FM2_Edition2 CER_ES4_PFM+FM1_Edition1 CER_ES4_PFM+FM2_Edition1 CER_ES4_TRMM-PFM_Edition1 CER_ES4_TRMM-PFM_Edition2 CER_ES4_Terra-FM1_Edition1 CER_ES4_Terra-FM2_Edition1 CER_ES4_FM1+FM2_Edition2 CER_ES4_Terra-FM1_Edition2 CER_ES4_Terra-FM2_Edition2 CER_ES4_Aqua-FM3_Edition1 CER_ES4_Aqua-FM4_Edition1 CER_ES4_FM1+FM2+FM3+FM4_Edition1 CER_ES4_Aqua-FM3_Edition2 CER_ES4_Aqua-FM4_Edition2 CER_ES4_FM1+FM3_Edition2 CER_ES4_FM1+FM4_Edition2 CER_ES4_PFM+FM1_Edition2 CER_ES4_PFM+FM2_Edition2 CER_ES4_Aqua-FM3_Edition1-CV CER_ES4_Aqua-FM4_Edition1-CV CER_ES4_Terra-FM1_Edition1-CV CER_ES4_Terra-FM2_Edition1-CV. [Location=GLOBAL] [Temporal_Coverage: Start_Date=1998-01-01; Stop_Date=2005-12-31] [Spatial_Coverage: Southernmost_Latitude=-90; Northernmost_Latitude=90; Westernmost_Longitude=-180; Easternmost_Longitude=180] [Data_Resolution: Latitude_Resolution=2.5 degree; Longitude_Resolution=2.5 degree; Horizontal_Resolution_Range=250 km - < 500 km or approximately 2.5 degrees - < 5.0 degrees; Temporal_Resolution=1 month; Temporal_Resolution_Range=Monthly - < Annual].

CLEAR-SKY ALBEDO↗

Ultra-High Operation Temperature SiC-matrix Solar Thermal Air Receiver (HOTSSTAR) enabled by additive manufacturing: Test Facility & Performance Evaluations

Solar Heat for Industrial Processes (SHIP) cavity receivers are capable of generating electricity or industrial process heat by absorbing thermal energy from solar radiation, focused on a small area. The concentration of solar radiation on the small area of the receiver enables the achievement of high temperatures (ranging from 400°C to 1,100°C) of a working fluid, thus making the SHIP technology thermodynamically comparable with conventional power plants. A volumetric receiver consists of a porous structure-generally made of silicon carbide or metal, which absorbs solar radiation and converts it into heat energy. Heat energy from the porous materials is then transferred to the fluid following through them. A volumetric receiver acts as a convective heat exchanger, transferring heat to the fluid through convection. Open-loop volumetric receivers work with air at atmospheric pressure and are suitable for single-cycle or multi-cycle energy plants. A Model Based Systems Engineering (MBSE) approach was used to develop a test bed at Sandia national Laboratories (SNL) capable of demonstrating an open-loop volumetric air receiver developed by General Electric Aerospace (GE Aerospace). This paper presents the development of the various MBSE methods, test bed, and testing operations for the GE air receiver, which was experimentally demonstrated to achieve 1,350°C for over 3 hours of operation and an approximate 70% receiver efficiency. By being able to achieve such high temperatures >1,000°C, this work provides the potential to support many SHIP industrial use cases.

14 SOLAR ENERGY↗

Black-Box Optimization for Design of Concentrating Solar Power and Photovoltaic Hybrid Systems with Optimal Dispatch Decisions

The hybridization of concentrating solar power (CSP) and photovoltaics (PV) can enable dispatchable renewable electricity generation at a lower price than current stand-alone CSP systems. However, designing a CSP-PV hybrid system can be challenging because of the many degrees of freedom in design that affect the internal and external system interactions and trade-offs. We develop a methodology to determine optimal designs for CSP-PV hybrids by implementing NLopt's derivative-free, or “black-box,” algorithms around pre-existing CSP-PV hybrid simulation software that utilizes the National Renewable Energy Laboratory’s System Advisor Model (SAM); we then employ a dispatch optimization model to determine operational decisions that maximize a plant’s profits. We present optimal designs for CSP-PV hybrid systems dispatching against four time-of-delivery (ToD) pricing structures. NLopt’s algorithms can improve the base case design’s power purchase agreement (PPA) price by 15% to 21%, depending on the ToD pricing structure. In addition, we present the resulting optimal CSP-PV hybrid design’s annual performance metrics, which tend to have capacity factors between 50% and 62%, but are able to generate electricity during the year’s highest-valued periods about 90% of the time. Lastly, we investigate the trade-offs between capacity factor and PPA price using Pareto fronts and demonstrate that, for some ToD pricing structures, the system capacity factor can increase by 20% but at the expense of a 2% increase in PPA price.

black-box↗

CalWave - Open Water Demonstration - System and Component Overview Content Models and Drawings

Data for the CalWave - Open Water Demonstration, a submerged pressure differential Wave Energy Converter (WEC) Device. Device is moored to the seabed, and the motion of the waves causes the sea level to rise and fall above the device, inducing a pressure differential in the device. The alternating pressure pumps fluid through a system to generate electricity, which is transmitted to shore via bidirectional cables. Documentation and data here includes: System Overview Content Model and Drawings as well as Component Overview Content Model and Drawings.

16 TIDAL AND WAVE POWER↗

Advancing Pumped Thermal Energy Storage Performance and Cost Using Silica Storage Media

Pumped Thermal Energy Storage (PTES) is an electricity storage system that is suitable for long-duration energy storage (10-1000 h) due to its low marginal cost of energy capacity. We present a techno-economic model of a PTES system that uses particle thermal energy storage. Particles have low costs and can be operated over a wide temperature range leading to increased efficiency and reduced costs compared to other PTES designs. We show how the round-trip efficiency, specific power output, capital cost, and levelized cost of storage (LCOS) depend on parameters such as the pressure ratio, heat exchanger approach temperature and pressure loss, and maximum temperature. We compare particle-PTES (P-PTES) performance to PTES which uses liquid thermal energy storage - i.e. molten salt hot storage and methanol cold storage (MS-PTES). We find that using silica particles for storage advances PTES technology: P-PTES can be operated at higher maximum temperatures than MS-PTES. Furthermore, P-PTES can achieve lower approach temperatures in the heat exchangers than MS-PTES without increasing capital costs, because P-PTES uses direct-contact heat transfer in fluidized bed heat exchangers. As a result, we find that P-PTES systems achieve higher round-trip efficiency than MS-PTES (66 % versus 57 %) and lower LCOS (e.g. 0.115 +- 0.03 $/kWhe versus 0.171 +- 0.04 $/kWhe for 10 h discharge). The low cost of particles and containment means that P-PTES can provide long-duration energy storage at low capital cost per unit energy capacity. For example, the total capital cost per unit energy reduces from 245 $/kWhe at 10 h to 38 $/kWhe at 100 h. These costs are considerably lower than MS-PTES (72 $/kWhe at 100 h) and also outcompete current and future Li-ion battery system projections (100-265 $/kWhe).

25 ENERGY STORAGE↗

Influence of the air-cooled supercritical carbon dioxide recompression cycle design-point on molten salt power tower techno-economic performance

Here, this paper investigates the techno-economic performance of a molten salt power tower integrated with an air-cooled supercritical carbon dioxide recompression cycle. Prior research into the cycle design point showed trade-offs between efficiency, cost, and temperature difference across the heat input, resulting in trade-offs between the design-point generation of the molten salt power tower system and the cost of sensible-heat thermal energy storage. Prior research into cycle off-design showed that colder ambient temperature designs have higher efficiencies on cold days but lower efficiency and capacity degradation on hot days. We present a framework to evaluate how cycle design, cost, and corresponding off-design performance affect the techno-economic performance of the molten salt power tower system. We use a steady-state off-design cycle model assuming fixed shaft speeds, inventory control, and air-cooler fan power control and integrate the cycle model with the molten salt power tower model from the National Renewable Energy Laboratory's System Advisor Model. First, we present a surrogate model that uses a heuristic approach to correct the interpolation when the cycle reaches capacity limitations at ambient temperatures hotter than design. Next, we evaluate how the recompression cycle design-point ambient temperature, air-cooler approach temperature, air-cooler fan power, and recuperator conductance influence cycle and system design metrics. Finally, we complete annual simulations using an hourly "duck curve" pricing schedule and present annual performance metrics to evaluate the cumulative effects of the cycle design parameters and corresponding off-design performance on molten salt power tower techno-economics.

14 SOLAR ENERGY↗

Climate Model Evaluation using New Datasets from the Clouds and the Earth's Radiant Energy System (CERES)

There are some in the science community who believe that the response of the climate system to anthropogenic radiative forcing is unpredictable and we should therefore call off the quest . The key limitation in climate predictability is associated with cloud feedback. Narrowing the uncertainty in cloud feedback (and therefore climate sensitivity) requires optimal use of the best available observations to evaluate and improve climate model processes and constrain climate model simulations over longer time scales. The Clouds and the Earth s Radiant Energy System (CERES) is a satellite-based program that provides global cloud, aerosol and radiative flux observations for improving our understanding of cloud-aerosol-radiation feedbacks in the Earth s climate system. CERES is the successor to the Earth Radiation Budget Experiment (ERBE), which has widely been used to evaluate climate models both at short time scales (e.g., process studies) and at decadal time scales. A CERES instrument flew on the TRMM satellite and captured the dramatic 1998 El Nino, and four other CERES instruments are currently flying aboard the Terra and Aqua platforms. Plans are underway to fly the remaining copy of CERES on the upcoming NPP spacecraft (mid-2010 launch date). Every aspect of CERES represents a significant improvement over ERBE. While both CERES and ERBE measure broadband radiation, CERES calibration is a factor of 2 better than ERBE. In order to improve the characterization of clouds and aerosols within a CERES footprint, we use coincident higher-resolution imager observations (VIRS, MODIS or VIIRS) to provide a consistent cloud-aerosol-radiation dataset at climate accuracy. Improved radiative fluxes are obtained by using new CERES-derived Angular Distribution Models (ADMs) for converting measured radiances to fluxes. CERES radiative fluxes are a factor of 2 more accurate than ERBE overall, but the improvement by cloud type and at high latitudes can be as high as a factor of 5. Diurnal cycles are explicitly resolved by merging geostationary satellite observations with CERES and MODIS. Atmospheric state data are provided from a frozen version of the Global Modeling and Assimilation Office- Data Assimilation System at the NASA Goddard Space Flight Center. In addition to improving the accuracy of top-of-atmosphere (TOA) radiative fluxes, CERES also produces radiative fluxes at the surface and at several levels in the atmosphere using radiative transfer modeling, constrained at the TOA by CERES (ERBE was limited to the TOA). In all, CERES uses 11 instruments on 7 spacecraft all integrated to obtain climate accuracy in TOA to surface fluxes. This presentation will provide an overview of several new CERES datasets of interest to the climate community (including a new adjusted TOA flux dataset constrained by estimates of heat storage in the Earth system), show direct comparisons between CERES ad ERBE, and provide a detailed error analysis of CERES fluxes at various time and space scales. We discuss how observations can be used to reduce uncertainties in cloud feedback and climate sensitivity and strongly argue why we should NOT "call off the quest".

Loeb, Norman G.↗

Experimental platforms for investigating feature-driven jets for HED mix model validation

High-energy-density (HED) systems, such as inertial confinement fusion (ICF), are susceptible to hydrodynamic instabilities that can significantly affect both experimental results and modeling predictions. Isolated features, such as fill tubes or divots in the capsule, can cause material to jet as a result of the compressive shock exciting the Richtmyer–Meshkov instability, and serve as one of the primary degradation mechanisms in ICF yield. Simulations of feature-driven jets and how they mix require extensive experimental validation, particularly for understanding to what degree the initial size and shape of a feature influence jet dynamics, and how much instability feeds through downstream layers. A better understanding of feature-driven jetting can improve our mix modeling capabilities and increase hydrodynamic simulation accuracy. This manuscript describes a series of experimental platforms fielded by Los Alamos National Laboratory as a part of the Mshock Omega 60 and ModCons Omega EP campaigns to explore feature-driven jetting. These platforms are designed to benchmark jet evolution and growth as a function of initial feature size and shape, investigate jet-layer interactions leading to instability feedthrough, and will be used to characterize jet-jet interactions resulting from clusters of features. In conclusion, preliminary results for both platforms are shown. The ModCons experiments are on-going, and a discussion of future work directions is included.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Evaluation of low-exergy heating and cooling systems and topology optimization for deep energy savings at the urban district level

District energy systems have the potential to achieve deep energy savings by leveraging the density and diversity of loads in urban districts. However, planning and adoption of district thermal energy systems is hindered by the analytical burden and high infrastructure costs. It is hypothesized that network topology optimization would enable wider adoption of advanced (ambient temperature) district thermal energy systems, resulting in energy savings. In this study, energy modeling is used to compare the energy performance of “conventional” and “advanced” district thermal energy systems at the urban district level, and a partial exhaustive search is used to evaluate a heuristic for the topology optimization problem. For the prototypical district considered, advanced district thermal energy systems mated with low-exergy building heating and cooling systems achieved a source energy use intensity that was 49% lower than that of conventional systems. The minimal spanning tree heuristic was demonstrated to be effective for the network topology optimization problem in the context of a prototypical district, and contributes to mitigating the problem’s computational complexity. The work presented in this paper demonstrates the potential of advanced district thermal energy systems to achieve deep energy savings, and advances to addressing barriers to their adoption through topology optimization.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

Surrogate Modeling for Capacity Planning of Charging Station Equipped with PV and Hydropneumatic Energy Storage

Due to the promising potential for environmental sustainability, there has been a significant increase of electric vehicles (EVs) and plug-in hybrid electric vehicles (PHEV) in the market. To support this increasing demand for EVs and PHEVs, challenges related to capacity planning and investment costs of public charging infrastructure must be addressed. Hence, in this paper, a capacity planning problem for EV charging stations is developed and aims to balance current capital investment costs and future operational revenue. The charging station considered in this work is assumed to be equipped with solar photovoltaic panel (PV) and an energy storage system which could be electric battery or the recently invented hydro-pneumatic energy storage (GLIDES, Ground-Level Integrated Diverse Energy Storage) system. A co-optimization model that minimizes investment and operation cost is established to determine the global optimal solution while combining the capacity and operational decision making. The operational decision making considers EV mobility which is modeled as an Erlang-loss system. Meanwhile, stochastic programming is adopted to capture uncertainties from solar radiation and charging demand of the EV fleet. To provide a more general and computationally efficient model, main configuration parameters are sampled in the design space and then fixed in solving the co-optimization model. The model can be used to provide insights for charging station placement in different practical situations. The sampled parameters include: the total number of EV charging slots, the PV area, the maximum capacity of the energy storage system, and daily mean EV arrival number in the Erlang-loss system. Based on the sampled parameter combinations and its responses, black-box mappings are then constructed using surrogate models (RBF, Kriging etc). The effectiveness of proposed surrogate modeling approach is demonstrated in the numerical experiments.

Chen, Yang↗

Including Physics-Informed Atomization Constraints in Neural Networks for Reactive Chemistry

Machine learning interatomic potentials (MLIPs) have emerged as powerful tools for investigating atomistic systems with high accuracy and a relatively low computational cost. However, a common and unaddressed challenge with many current neural network (NN) MLIP models is their limited ability to accurately predict the relative energies of systems containing isolated or nearly isolated atoms, which appear in various reactive processes. To address this limitation, we present a mathematical technique for modifying any existing atom-centered NN architecture to account for the energies of isolated atoms. The result produces a consistent prediction of the atomization energy (AE) of a system using minimal constraints on the model. Using this technique, we build a model architecture that we call hierarchically interacting particle neural network (HIP-NN)-AE, an AE-constrained version of the HIP-NN, as well as ANI-AE, the AE-constrained version of the accurate NN engine for molecular energies (ANI). Our results demonstrate AE consistency of AE-constrained models, which drastically improves the AE predictions for the models. We compare the AE-constrained approach to unconstrained models as well as models from the literature in other scenarios, such as bond dissociation energies, bond dissociation pathways, and extensibility tests. These results show that the constraints improve the model performance in some of these tasks and do not negatively affect the performance on any tasks. The AE constraint approach thus offers a robust solution to the challenges posed by isolated atoms in energy prediction tasks.

74 ATOMIC AND MOLECULAR PHYSICS↗

Revenue prediction for integrated renewable energy and energy storage system using machine learning techniques

Revenue estimation for integrated renewable energy and energy storage systems is important to support plant owners or operators’ decisions in battery sizing selection that leads to maximized financial performances. A common approach to optimizing revenues of a hybrid hydro and energy storage system is using mixed-integer linear programming (MILP). Although MILP models can provide accurate production cost estimations, they are typically very computationally expensive. To provide a fast yet accurate first-step information to hydropower plant owners or operators who consider integrating energy storage systems, we propose an innovative approach to predicting optimal revenues of an integrated energy generation and storage system. In this study, we examined the performance of two prediction techniques: Generalized Additive Models (GAMs) and machine learning (ML) models developed based on artificial neural networks (ANN). Predictive equations and models are generated based on optimized solutions from a market participation optimization model, the Conventional Hydropower Energy and Environmental Resource System (CHEERS) model. The two predicting techniques reduce the computational time to evaluate annual revenue for one set of battery configurations from 3 h to 1 to 4 min per run while also being implementable with significantly less data. The model validation prediction errors of developed GAMs and ML models are generally below 5%; for model testing predictions, the ML models consistently outperform the regression equations in terms of root mean square errors. This new approach allows plant owners, operators, or potential investors to quickly access multiple battery configurations under different energy generation and market scenarios. This new revenue prediction method will therefore help reduce the barriers, and thereby promoting the deployment of battery hybridization with existing renewable energy sources.

13 HYDRO ENERGY↗

Valley Hall Effect and Magnetic Moment in Magnetized Silicene

We investigate the physical properties of silicene with both staggered sublattice potential and magnetization by using Kubo formalism, the latter arises from the magnetic proximity effect by depositing Fe atoms to silicene or depositing silicene on an appropriate ferromagnetic insulator. Based on the low-energy continuum model of the system where inversion symmetry is broken, we show that the system exhibits spin half metal state when staggered sublattice potential is in the same magnitude with mean and staggered magnetization. Besides, Hall conductivity and magnetic moment are all valley dependent, so we investigate the valley Hall effect of the system further by considering magnetization exclusively. This means carriers in different valleys turning into opposite directions transverse to an in-plane electric field. At last, we prove these results by investigating Berry curvature that characterizing Hall transport, which is also valley dependent. These effects can be used to generate valley-polarized currents solely by magnetization, forming the basis for the valley-based electronics applications.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

Development of typical solar years and typical wind years for efficient assessment of renewable energy systems across the U.S.

Weather data plays a critical role in renewable energy analysis. Compared to using multiple Actual Meteorological Years, simulations using a single typical year require significantly fewer computational resources. Previous efforts to create typical weather datasets for renewable energy analysis either lack justified or optimized strategies for selecting and weighting different weather parameters or are limited to a few specific locations. Here, in this study, we developed a dataset comprising Typical Solar Years (TSYs) and Typical Wind Years (TWYs) for over 2000 locations across the U.S., based on data from NASA's POWER project. The strategies for creating TSYs and TWYs were optimized based on the simulated outputs of various PV systems and wind turbines in 16 representative cities. This dataset provides an efficient means for the rapid evaluation and optimization of renewable energy systems throughout the entire U.S. Additionally, the optimal strategies identified in this study can be directly applied to create near-optimal TSYs and TWYs for most locations worldwide. However, readers can also employ the optimization approach presented in this work to develop optimal strategies tailored for particular regions.

NASA POWER↗

CASM Monte Carlo: Calculations of the thermodynamic and kinetic properties of complex multicomponent crystals

Monte Carlo techniques play a central role in statistical mechanics approaches that connect macroscopic thermodynamic and kinetic properties to the electronic structure of a material. This paper describes the implementation of Monte Carlo techniques for the study of multicomponent crystalline materials within the Clusters Approach to Statistical Mechanics (CASM) software suite, and demonstrates their use in model systems to calculate free energies and kinetic coefficients, study phase transitions, and construct phase diagrams from first principles. Many crystal structures are complex, with multiple sublattices occupied by differing sets of chemical species, along with the presence of vacancies or interstitial species. This imposes constraints on concentration variables, the form of thermodynamic potentials, and the values of kinetic transport coefficients. The framework used by CASM to formulate thermodynamic potentials and kinetic transport coefficients accounting for arbitrarily complex crystal structures is presented and demonstrated with examples of increasing complexity. Additionally, an overview of the capabilities of the CASM software specific to Monte Carlo methods is given, and a new CASM software package is introduced, casm-flow, which helps automate the setup, submission, management, and analysis of Monte Carlo simulations.

Cluster expansion↗