Results for “Illinois” — Page 42

NASA NTRS2022

Experimental and Computational Icing Simulation for Large Swept Wings

The purpose of this final report is to summarize the key results, findings and contributions of a large, multi-organization, multi-year effort focused on swept wing icing and aerodynamics. This research effort was jointly sponsored by NASA, FAA and ONERA and was supported by Boeing and the Universities of Illinois, Virginia, and Washington. The overall goal of this research was to improve the experimental and computational simulation capability for icing on large swept wings typical of commercial transports. This research included both ice accretion and aerodynamic studies using the NASA Common Research Model (CRM) as the reference geometry. For this work, a 65% scaled version—CRM65—was used as the full-scale baseline airplane geometry. This program marks the first non-proprietary research to generate and document full-scale swept wing ice accretions in an icing tunnel and then conduct semispan wing aerodynamic testing on scaled ice accretions at high Reynolds number (i.e., up to 11.9×106). During this work, several new experimental and computational techniques were developed or documented and new knowledge about swept wing icing aerodynamics was identified. All are documented in this report.

Aircraft icing↗

NASA NTRS2021

Low-Maintenance Bioreactor Cultivates Fungi for Sustainable Food Source in Space

Nature’s Fynd, a food company headquartered in Chicago, Illinois, partnered with Montana State University on NASA STTR contracts to further develop a micro-gravity biofilm-biomat reactor, which cultivates a unique fungus to form a dense protein material. The resulting “biomat” could serve as a nutritious food source for life away from Earth. Nature’s Fynd has received external investments totaling more than $500 million for developing its technologies, and the company recently launched its meatless and dairy-free foods in specific retailers.

STTR↗

NASA NTRSDate not supplied

Expanding the Spatiotemporal Range of Soil Moisture Analysis Utilizing NASA Earth Observations and In Situ Measurements

Drought can cause immense agricultural and ecological damage resulting in high mitigation and compensation costs. Climate variability in future decades is expected to cause severe drought conditions and threaten necessary water resources. Stakeholders seek to implement effective drought assessments in preparation for potential economic and environmental damage invoked by drought. Although in-situ measurements are accurate, the current infrastructure is spatially limited and costly to maintain. A framework was created to compare modeled, satellite and in-situ data in drought monitoring. Here we show that the comparison of in-situ and remotely sensed soil moisture (SM) measurements can increase the spatiotemporal range of SM assessments. Data collected between 2003 and 2021 by NASA’s SPoRT Land Information System (SPoRT-LIS) and Soil Moisture Active Passive (SMAP) mission were standardized and compared with in-situ data provided through the Illinois Climate Network (WARM). Statistical analysis results including the Pearson correlation coefficient (r), root mean squared error, mean absolute error and others were calculated to compare the WARM measurements to the SMAP and SPoRT-LIS products. Results indicate that both satellite products demonstrate seasonally variable bias that is not present in the in-situ measurements. Bias was highest in the winter months and lowest in the late summer and early fall months in both satellite datasets. Overall, WARM-SPoRT comparisons resulted in lower seasonal variability. However, on average, the SMAP comparison demonstrated higher correlation values and lower error values. The WARM-SMAP average correlation (r) was 0.61 compared to the WARM-SPoRT average correlation (r) value of 0.54. Average mean absolute error values calculated for the SMAP and SPoRT comparisons were 0.07 and 0.08 percent soil moisture by volume, respectively. These analyses suggest integrating in-situ measurements and those provided by NASA Earth observations can be utilized in a multi-faceted SM evaluation, a valuable contribution to drought monitoring and water resource decision making.

Emma P Myrick↗

NASA NTRS2022

PATOKA: Simulating Electromagnetic Observables of Black Hole Accretion

The Event Horizon Telescope (EHT) has released analyses of reconstructed images of horizon-scale millimeter emission near the supermassive black hole at the center of the M87 galaxy. Parts of the analyses made use of a large library of synthetic black hole images and spectra, which were produced using numerical general relativistic magnetohydrodynamics fluid simulations and polarized ray tracing. In this article, we describe the PATOKA pipeline, which was used to generate the Illinois contribution to the EHT simulation library. We begin by describing the relevant accretion systems and radiative processes. We then describe the details of the three numerical codes we use, iharm, ipole, and igrmonty, paying particular attention to differences between the current generation of the codes and the originally published versions. Finally, we provide a brief overview of simulated data as produced by PATOKA and conclude with a discussion of limitations and future directions.

supermassive black holes↗

NASA NTRS2021

Hybrid phenology matching model for robust crop phenological retrieval

Crop phenology regulates seasonal agroecosystem carbon, water, and energy exchanges, and is a key component in empirical and process-based crop models for simulating biogeochemical cycles of farmlands, assessing gross and net primary production, and forecasting the crop yield. The advances in phenology matching models provide a feasible means to monitor crop phenological progress using remote sensing observations, with a priori information of reference shapes and reference phenological transition dates. Yet the underlying geometrical scaling assumption of models, together with the challenge in defining phenological references, hinders the applicability of phenology matching in crop phenological studies. The objective of this study is to develop a novel hybrid phenology matching model to robustly retrieve a diverse spectrum of crop phenological stages using satellite time series. The devised hybrid model leverages the complementary strengths of phenometric extraction methods and phenology matching models. It relaxes the geometrical scaling assumption and can characterize key phenological stages of crop cycles, ranging from farming practice-relevant stages (e.g., planted and harvested) to crop development stages (e.g., emerged and mature). To systematically evaluate the influence of phenological references on phenology matching, four representative phenological reference scenarios under varying levels of phenological calibrations in terms of time and space are further designed with publicly accessible phenological information. The results indicate that the hybrid phenology matching model can achieve high accuracies for estimating corn and soybean phenological growth stages in Illinois, particularly with the year- and region-adjusted phenological reference (R-squared higher than 0.9 and RMSE less than 5 days for most phenological stages). The inter-annual and regional phenological patterns characterized by the hybrid model correspond well with those in the crop progress reports (CPRs) from the USDA National Agricultural Statistics Service (NASS). Compared to the benchmark phenology matching model, the hybrid model is more robust to the decreasing levels of phenological reference calibrations, and is particularly advantageous in retrieving crop early phenological stages (e.g., planted and emerged stages) when the phenological reference information is limited. This innovative hybrid phenology matching model, together with CPR-enabled phenological reference calibrations, holds 3 considerable promise in revealing spatio-temporal patterns of crop phenology over extended geographical regions.

Phenology↗

NASA NTRSDate not supplied

Modeling instrumented test articles for Plasmatron X aerothermal environment

This work discusses the ablative response of thermal protection system materials when exposed to aerothermal environments. It focuses on modeling instrumented test article assemblies tested in the Plasmatron X inductively coupled plasma wind tunnel. For this purpose, as the samples are mounted to a water-cooled arm using numerous assembly components and are embedded with thermocouples for in-depth temperature measurements, the variation in ablative response of the samples as a result of these components is studied. The aerothermal boundary conditions for Plasmatron X are obtained for hypersonic flight-relevant conditions using the multi-physics aerothermal framework built at the University of Illinois at Urbana-Champaign. For simulating the material response, a material response code, Porous Material Analysis Toolbox based on OpenFOAM (PATO) is used. In PATO, numerous configurations and materials of thermocouples were studied. The optimal configuration for obtaining temperature measurements with minimal error using embedded thermocouples was found to be the U-shaped Type S, 30 American wire gauge thermocouple configuration.

Thermal Protection Systems↗

NASA NTRSDate not supplied

Modeling instrumented test articles for Plasmatron X aerothermal environment

This work discusses the ablative response of thermal protection system materials when exposed to aerothermal environments. It focuses on modeling instrumented test article assemblies tested in the Plasmatron X inductively coupled plasma wind tunnel. For this purpose, as the samples are mounted to a water-cooled arm using numerous assembly components and are embedded with thermocouples for in-depth temperature measurements, the variation in ablative response of the samples as a result of these components is studied. The aerothermal boundary conditions for Plasmatron X are obtained for hypersonic flight-relevant conditions using the multi-physics aerothermal framework built at the University of Illinois at Urbana-Champaign. For simulating the material response, a material response code, Porous Material Analysis Toolbox based on OpenFOAM (PATO) is used. In PATO, numerous configurations and materials of thermocouples were studied. The optimal configuration for obtaining temperature measurements with minimal error using embedded thermocouples was found to be the U-shaped Type S, 30 American wire gauge thermocouple configuration.

Thermal Protection Systems↗

NASA NTRS2023

Kankakee Water Resources: Monitoring Temperature and Vegetation to Detect River Flow Impediments at Energy Intake Structures

In recent years, unpredictable grassing events have occurred at the Dresden Generating Station, located on the Kankakee River in northern Illinois. Grassing events are characterized by large mats of aquatic vegetation that accumulate downstream, resulting in the clogging of water intake structures and leading to major disruptions in power generation. Currently, employees at the Dresden Generating Station are responsible for reactively responding to each grassing event individually. This project, in partnership with Constellation Nuclear and the United States Geological Survey (USGS), assessed the feasibility of using Earth observations (Landsat 9 OLI-2, Landsat 8 OLI, Sentinel-2 MSI, DOVE PlanetScope, WorldView-3, and GPM IMERG) to detect floating aquatic vegetation within the Kankakee River and identify predictive factors that trigger grassing events, as doing so will provide the Dresden Generating Station the ability to anticipate future grassing events and enhance general hydrologic modeling efforts held by the USGS. The results of this study illustrated that, while aquatic vegetation can be detected by satellites with up to moderate spatial resolution (30 m), temporal resolution is a major limiting factor for tracking movements in floating aquatic vegetation and identifying predictive measures for these events. In addition, correlation results suggest a possible negative relationship between grassing events and river discharge (-0.875 correlation coefficient). In the future, pairing these results with ground control surveys and sensors with higher temporal capabilities would allow our project partners to predict and proactively address future grassing events, ensuring the reliable operation of the Dresden Generating Station.

Marisa Smedsrud↗

NASA NTRSDate not supplied

High-Enthalpy Testing to Validate Simulation of an Aerosol Capture Probe

AERACEPT (Aerosol Rapid Analysis Combined Entry Probe/sonde Technology) is a developing technology enabling in-situ aerosol particle sampling and analysis in a small spacecraft mission envelope. It integrates a passive aerosol sample collection system into a probe’s thermal protection system (TPS) to remove the need for heat shield separation and active descent control (parachutes, gliders, etc.). AERACEPT is being validated against the requirements of the Nephele mission concept, which targets the middle and lower Venus cloud layers. The proposed technology employs 3D Carbon-Carbon (3D-CC) at the probe’s nose to withstand the extreme environments encountered in a Venus entry without producing pyrolysis gases that would contaminate the mission sample collection. Any shape change of this 3D-CC sample inlet will affect airflow through the sampling system, thereby impacting aerosol sample collection bias and efficiency. Accurate prediction of the nose’s material response during entry is there-fore of fundamental importance to the development and practical use of this technology. As a result, the AERACEPT project is planning a test campaign in the newly built PlasmatronX facility with a novel “Open Iso-Q” test article to validate state-of-the-art material response tools for a simulated Venus entry. The PlasmatronX is a 350 kW inductively-coupled plasma facility developed and run by the Center for Hypersonics and Entry Systems Studies at the University of Illinois at Urbana-Champaign. The facility can support ground testing for a variety of planetary destinations by simulating entry conditions in Nitrogen, Air, and Carbon Dioxide. To best simulate the Venus atmosphere and match flight-like recession, the planned AERACEPT test campaign will use Carbon Dioxide as its test gas. AERACEPT has designed two test articles for this campaign: 1) a standard test article whose curved surface approximates a constant applied heat flux (“Closed Iso-Q” model) and 2) a novel inlet test article with a through-hole at the stagnation point (“Open Iso-Q” model). Both article types consist of a 3D-CC sample bonded to a graphite fixture. All articles will be laser scanned before and after testing to estimate shape change and recession, and the “Closed Iso-Q” articles will be instrumented with thermocouples to provide temperature histories. The test article temperatures and inlet shape change will be compared against simulations run with the Porous material Analysis Toolbox based on OpenFOAM (PATO) to validate the AERACEPT 3D-CC material response model. Due to the high thermal conductivity of 3D-CC, multidimensional effects have a large influence on the test article temperature and the standard 1-D material response tools are not sufficient for test planning. The test campaign features (e.g. article design, run duration, heat fluxes) are therefore informed by a series of 3-D simulations using state-of-the-art material response tools such as PATO. The test campaign is scheduled for spring of 2024. Expected results include 1) Key drivers of selected test conditions (e.g. matching mission flight atmosphere & recession), 2) Simulation results that informed test article design, 3) Acquired test data such as recession measurements, temperature histories, and photos, and 4) Comparisons between test results and simulation predictions.

AERACEPT↗

NASA NTRS2024

Flexural Fatigue Testing and Qualification of Hybrid PAN-pitch Composite Materials with High Through-Thickness Thermal Conductivity

Carbon-fiber-reinforced polymers (CFRPs) considered for motor components within future electric vertical takeoff and landing (eVTOL) vehicles require not only a high strength-to-weight ratio but also a high through-thickness thermal conductivity (TC) to dissipate heat. Previously, CFRPs incorporating pitch CF interleaves demonstrated an increased through-thickness TC relative to the baseline composite, but the effect on flexural fatigue life was not evaluated. For this work, T700S/TC275–1 braided carbon fiber/epoxy prepreg baseline (no interleave), YSH–60A pitch interleave, and YS–80A pitch interleave configurations were tested for fatigue life. A combination of ASTM International Standards D7264 and D3479 were used to develop a reliable testing method. An Instron® (Illinois Tool Works Inc.) 8501 servohydraulic system with a fatigue test fixture was set up with a 16:1 span:thickness ratio, load ratio R set to 0.1, and frequency of 4 Hz. High-cycle fatigue testing was completed, and it was found that YSH–60A interleave performed slightly better than YS–80A below 100,000 cycles but showed no differences above 100,000 cycles. In all cases, the baseline CFRP had superior fatigue life performance both in terms of stress handled and total cycles to failure. These fatigue testing data will inform future use of hybrid CFRP materials by highlighting the tradeoff between increased through-thickness TC and fatigue life. Testing guidelines, equipment, and parameters for static and fatigue tests were successfully outlined for CFRPs.

composites↗

NASA NTRSDate not supplied

High-Enthalpy Testing to Validate Simulation of an Aerosol Capture Probe

AERACEPT (Aerosol Rapid Analysis Combined Entry Probe/sonde Technology) is a developing technology enabling in-situ aerosol particle sampling and analysis in a small spacecraft mission envelope. It integrates a passive aerosol sample collection system into a probe’s thermal protection system (TPS) to remove the need for heat shield separation and active descent control (parachutes, gliders, etc.). AERACEPT is being validated against the requirements of the Nephele mission concept, which targets the middle and lower Venus cloud layers. The proposed technology employs 3D Carbon-Carbon (3D-CC) at the probe’s nose to withstand the extreme environments encountered in a Venus entry without producing pyrolysis gases that would contaminate the mission sample collection. Any shape change of this 3D-CC sample inlet will affect airflow through the sampling system, thereby impacting aerosol sample collection bias and efficiency. Accurate prediction of the nose’s material response during entry is there-fore of fundamental importance to the development and practical use of this technology. As a result, the AERACEPT project is planning a test campaign in the newly built PlasmatronX facility with a novel “Open Iso-Q” test article to validate state-of-the-art material response tools for a simulated Venus entry. The PlasmatronX is a 350 kW inductively-coupled plasma facility developed and run by the Center for Hypersonics and Entry Systems Studies at the University of Illinois at Urbana-Champaign. The facility can support ground testing for a variety of planetary destinations by simulating entry conditions in Nitrogen, Air, and Carbon Dioxide. To best simulate the Venus atmosphere and match flight-like recession, the planned AERACEPT test campaign will use Carbon Dioxide as its test gas. AERACEPT has designed two test articles for this campaign: 1) a standard test article whose curved surface approximates a constant applied heat flux (“Closed Iso-Q” model) and 2) a novel inlet test article with a through-hole at the stagnation point (“Open Iso-Q” model). Both article types consist of a 3D-CC sample bonded to a graphite fixture. All articles will be laser scanned before and after testing to estimate shape change and recession, and the “Closed Iso-Q” articles will be instrumented with thermocouples to provide temperature histories. The test article temperatures and inlet shape change will be compared against simulations run with the Porous material Analysis Toolbox based on OpenFOAM (PATO) to validate the AERACEPT 3D-CC material response model. Due to the high thermal conductivity of 3D-CC, multidimensional effects have a large influence on the test article temperature and the standard 1-D material response tools are not sufficient for test planning. The test campaign features (e.g. article design, run duration, heat fluxes) are therefore informed by a series of 3-D simulations using state-of-the-art material response tools such as PATO. The test campaign is scheduled for spring of 2024. Expected results include 1) Key drivers of selected test conditions (e.g. matching mission flight atmosphere & recession), 2) Simulation results that informed test article design, 3) Acquired test data such as recession measurements, temperature histories, and photos, and 4) Comparisons between test results and simulation predictions.

AERACEPT↗

NASA NTRS2024

Toward 2D Dynamo Models Calibrated by Global 3D Relativistic Accretion Disk Simulations

Two-dimensional models assuming axisymmetry are an economical way to explore the long-term evolution of black hole accretion disks, but they are only realistic if the feedback of the nonaxisymmetric turbulence on the mean momentum and magnetic fields is incorporated. Dynamo terms added to the 2D induction equation should be calibrated to 3D MHD simulations. For generality, the dynamo tensors should be calibrated as functions of local variables rather than explicit functions of spatial coordinates in a particular basis. In this paper, we study the feedback of non-axisymmetric features on the 2D mean fields using a global 3D, relativistic, Cartesian simulation from the Illinois GRMHD code. We introduce new methods for estimating overall dynamo alpha and turbulent diffusivity effects as well as measures of the dominance of non-axisymmetric components of energies and fluxes within the disk interior. We attempt closure models of the dynamo EMF using least squares fitting, considering both models where coefficient tensors are functions of space and more global, covariant models. None of these models are judged satisfactory, but we are able to draw conclusions on what sorts of generalizations are and are not promising.

Matthew D Duez↗

NASA NTRSDate not supplied

Assessing the Feasibility of Using Various Earth Observations to Monitor Environmental Trends Associated with River Flow Impediments Near Energy Intake Structures

Grassing events, characterized by the release of river vegetation in large quantities, are often responsible for major flow impediments surrounding industrial water intake structures. These sudden flow impediments halt critical municipal functions, such as cooling at energy generating facilities, leading to widespread energy disruptions for surrounding communities. Because the origin and cause of grassing events in rivers are largely unknown, employees at these facilities can only reactively respond after they occur by pausing energy generation to manually remove the accumulation of aquatic vegetation. In 2023, the Dresden Generating Station, in collaboration with the United States Geological Survey, began looking into using remote sensing methodologies to locate floating aquatic vegetation surrounding their intake structure located along the Kankakee River in northern Illinois. This project contributed to this effort by conducting a case study that compared the performance, practicality, and feasibility of using various Earth observations (Landsat 9 OLI-2, Landsat 8 OLI, Sentinel-2 MSI, and DOVE PlanetScope) and vegetation indices (NDVI, EVI, SAVI, and GCI) to monitor aquatic vegetation in the Kankakee River. Additionally, this study incorporated several environmental metrics to identify potential triggers for grassing events. Results from this project found that while Landsat sensors provide a lower spatial resolution than commercial satellite Earth observations, areas of aquatic vegetation were similarly identified; therefore, it may not be necessary to acquire expensive, high-resolution datasets for continued monitoring. Finally, preliminary correlation analyses showed a potential negative relationship between the river discharge and the presence of aquatic vegetation (-0.875 correlation coefficient), suggesting that periods of low flow could lead to large releases in aquatic vegetation.

Marisa Smedsrud↗

NASA NTRS2026

PROTECT: Production and Reuse of Thermally Efficient Ceramic Thermal Protection Systems

PROTECT (Production and Reuse Of Thermally Efficient Ceramic TPS) is a NASA Early Career Initiative focused on developing the next generation of reusable ceramic thermal protection systems (TPS). This project addresses key challenges in TPS design, including temperature capability, thermal stability, and refurbishment time, by integrating novel material development with predictive modeling. Leveraging enhanced capabilities in NASA’s Porous Microstructure Analysis (PuMA) software, PROTECT introduces new modeling tools to predict the thermal and mechanical behavior of fibrous ceramic materials. These tools enable accurate prediction of performance metrics such as thermal conductivity and structural integrity, reducing reliance on costly physical testing. Preliminary advances in these areas will be presented. To support model validation, PROTECT is building a comprehensive database of raw material properties using advanced characterization techniques, including micro computed tomography (CT) scanning in collaboration with the University of Illinois Urbana-Champaign (UIUC). The presentation will detail the sampling workflows and analysis methods used to generate this detailed microstructural data and how it is used to develop improved models in PuMA. This multi-center collaboration, spanning NASA (JSC, ARC, KSC, GRC), Oak Ridge National Laboratory, UIUC, and SpaceX, is developing tailored TPS solutions for LEO, lunar, and Martian missions. By bridging heritage systems with the demands of modern spaceflight, PROTECT contributes to the advancement of reusable TPS technologies for future exploration missions.

Propulsion, Refractory, and Coating Materials↗

NASA NTRS2026

PROTECT: Production and Reuse of Thermally Efficient Ceramic Thermal Protection Systems

PROTECT (Production and Reuse Of Thermally Efficient Ceramic TPS) is a NASA Early Career Initiative focused on developing the next generation of reusable ceramic thermal protection systems (TPS). This project addresses key challenges in TPS design, including temperature capability, thermal stability, and refurbishment time, by integrating novel material development with predictive modeling. Leveraging enhanced capabilities in NASA’s Porous Microstructure Analysis (PuMA) software, PROTECT introduces new modeling tools to predict the thermal and mechanical behavior of fibrous ceramic materials. These tools enable accurate prediction of performance metrics such as thermal conductivity and structural integrity, reducing reliance on costly physical testing. Preliminary advances in these areas will be presented. To support model validation, PROTECT is building a comprehensive database of raw material properties using advanced characterization techniques, including micro computed tomography (CT) scanning in collaboration with the University of Illinois Urbana-Champaign (UIUC). The presentation will detail the sampling workflows and analysis methods used to generate this detailed microstructural data and how it is used to develop improved models in PuMA. This multi-center collaboration, spanning NASA (JSC, ARC, KSC, GRC), Oak Ridge National Laboratory, UIUC, and SpaceX, is developing tailored TPS solutions for LEO, lunar, and Martian missions. By bridging heritage systems with the demands of modern spaceflight, PROTECT contributes to the advancement of reusable TPS technologies for future exploration missions.

Propulsion, Refractory, and Coating Materials↗

DOE OSTI2019

Multi-layered Resilient Microgrid Networks

Distribution grid resilience, especially when the grid is under high penetration of Distributed Energy Resources (DER), e.g. PV and energy storage, is essential to ensure the quality of electric power services. The resiliency aspect is particularly important for critical loads that typically include hospitals, police/fire stations, and national security facilities. Networks of multiple interconnected reconfigurable microgrids help to increase renewable energy penetration and may be used as basic building blocks for implementing future resilient power grids. However, this integration requires additional layers of communications, both horizontal and vertical, which may substantially increase the potential cyber attack surfaces as well as escalate risks for wide-spread malfunctions of automation systems. The Mulit-layered Resilient Microgrid Networks project is an industry-academic collaboration between ABB, Duke Energy and the University of Illinois at Urbana-Champaign (UIUC) that has developed and demonstrated a cyber-physical resilient control and communication architecture for deployment of multiple microgrid networks, as well as retrofitting the existing distribution grids by increasing DER penetration levels.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2020

Cyber-physical Resilient Interoperable Microgrid Networks

The paper presents the results of an industry-academic collaboration between ABB, Duke Energy and the University of Illinois at Urbana-Champaign (UIUC) that has developed and demonstrated a cyberphysical resilient control and communication architecture for deployment of multiple microgrid networks, as well as retrofitting the existing distribution grids by increasing DER penetration levels.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2020

Formulations and Valid Inequalities for Optimal Black Start Allocation in Power Systems

The restoration of a power system after a blackout starts around units with enhanced technical capabilities, referred to as black start units (BSUs). We examine the planning problem of optimally allocating these units on the grid subject to a budget constraint. We present a mixed integer programming model based on current literature in power systems. Binary variables are associated with the allocation of BSUs and with the energization state of buses, branches, and generators of the power system over a time horizon. We extract a substructure of the feasible region which imposes the requirement that each island that appears during the restoration process must have at least one operational generator. We discuss three equivalent reformulations for this requirement. We introduce a family of exponentially many, polynomially separable, valid inequalities to strengthen the formulation. Under simplifying assumptions, we show that the convex hull of the feasible region is a full-dimensional polyhedron, and prove that some of the constraints we introduced are facet-defining. We perform experiments to examine the difference in strength between the formulations as well as the computational times to solve the problem to near optimality for synthetic instances of the IEEE-39, IEEE-118, Illinois-200, WECC-225, IEEE-300, South Carolina-500, and Texas-2000 power systems. We illustrate a use case of the model. We conclude by suggesting extensions of the current work for future research.

Black Start Allocation↗