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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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All sourcesNASA NTRSDOE OSTI
NASA NTRS2010

Wavefront Control and Image Restoration with Less Computing

PseudoDiversity is a method of recovering the wavefront in a sparse- or segmented- aperture optical system typified by an interferometer or a telescope equipped with an adaptive primary mirror consisting of controllably slightly moveable segments. (PseudoDiversity should not be confused with a radio-antenna-arraying method called pseudodiversity.) As in the cases of other wavefront- recovery methods, the streams of wavefront data generated by means of PseudoDiversity are used as feedback signals for controlling electromechanical actuators of the various segments so as to correct wavefront errors and thereby, for example, obtain a clearer, steadier image of a distant object in the presence of atmospheric turbulence. There are numerous potential applications in astronomy, remote sensing from aircraft and spacecraft, targeting missiles, sighting military targets, and medical imaging (including microscopy) through such intervening media as cells or water. In comparison with prior wavefront-recovery methods used in adaptive optics, PseudoDiversity involves considerably simpler equipment and procedures and less computation. For PseudoDiversity, there is no need to install separate metrological equipment or to use any optomechanical components beyond those that are already parts of the optical system to which the method is applied. In Pseudo- Diversity, the actuators of a subset of the segments or subapertures are driven to make the segments dither in the piston, tilt, and tip degrees of freedom. Each aperture is dithered at a unique frequency at an amplitude of a half wavelength of light. During the dithering, images on the focal plane are detected and digitized at a rate of at least four samples per dither period. In the processing of the image samples, the use of different dither frequencies makes it possible to determine the separate effects of the various dithered segments or apertures. The digitized image-detector outputs are processed in the spatial-frequency (Fourier-transform) domain to obtain measures of the piston, tip, and tilt errors over each segment or subaperture. Once these measures are known, they are fed back to the actuators to correct the errors. In addition, measures of errors that remain after correction by use of the actuators are further utilized in an algorithm in which the image is phase-corrected in the spatial-frequency domain and then transformed back to the spatial domain at each time step and summed with the images from all previous time steps to obtain a final image having a greater signal-to-noise ratio (and, hence, a visual quality) higher than would otherwise be attainable.

Lyon, Richard G.↗
NASA NTRS2008

Orbit Determination During Spacecraft Emergencies with Sparse Tracking Data - THEMIS and TDRS-3 Lessons Learned

This paper provides an overview of the lessons learned from the National Aeronautics and Space Administration (NASA) Goddard Space Flight Center s (GSFC) Flight Dynamics Facility s (FDF) support of the Time History of Events and Macroscale Interactions during Substorms (THEMIS) spacecraft emergency in February 2007, and the Tracking and Data Relay Satellite-3 (TDRS-3) spacecraft emergency in March 2006. A successful and timely recovery from both of these spacecraft emergencies depended on accurate knowledge of the orbit. Unfortunately, the combination of each spacecraft emergency with very little tracking data contributed to difficulties in estimating and predicting the orbit and delayed recovery efforts in both cases. In both the THEMIS and TDRS-3 spacecraft emergencies, numerous factors contributed to problems with obtaining nominal tracking data measurements. This paper details the various causative factors and challenges. This paper further enumerates lessons learned from FDF s recovery efforts involving the THEMIS and TDRS-3 spacecraft emergencies and scant tracking data, as well as recommendations for improvements and corrective actions. In addition, this paper describes the broad range of resources and complex navigation methods employed within the FDF for supporting critical navigation activities during all mission phases, including launch, early orbit, and on-orbit operations.

Morinelli, Patrick J.↗
DOE OSTI2022

Quantifying Dynamic Signal Spread in Real-Time High-Energy X-ray Diffraction

Abstract Measured intensity in high-energy monochromatic X-ray diffraction (HEXD) experiments provides information regarding the microstructure of the crystalline material under study. The location of intensity on an areal detector is determined by the lattice spacing and orientation of crystals so that changes in the heterogeneity of these quantities are reflected in the spreading of diffraction peaks over time. High temporal resolution of such dynamics can now be experimentally observed using technologies such as the mixed-mode pixel array detector (MM-PAD) which facilitates in situ dynamic HEXD experiments to study plasticity and its underlying mechanisms. In this paper, we define and demonstrate a feature computed directly from such diffraction time series data quantifying signal spread in a manner that is correlated with plastic deformation of the sample. A distinguishing characteristic of the analysis is the capability to describe the evolution from the distinct diffraction peaks of an undeformed alloy sample through to the non-uniform Debye–Scherrer rings developed upon significant plastic deformation. We build on our previous work modeling data using an overcomplete dictionary by treating temporal measurements jointly to improve signal spread recovery. We demonstrate our approach in simulations and on experimental HEXD measurements captured using the MM-PAD. Our method for characterizing the temporal evolution of signal spread is shown to provide an informative means of data analysis that adds to the capabilities of existing methods. Our work draws on ideas from convolutional sparse coding and requires solving a coupled convex optimization problem based on the alternating direction method of multipliers.

Engineering↗
DOE OSTI2021

Attack-Resilient Weighted $\ell_{1}$ Observer with Prior Pruning

Security related questions for Cyber Physical Systems (CPS) have attracted much research attention in searching for novel methods for attack-resilient control and/or estimation. Specifically, false data injection attacks (FDIAs) have been shown to be capable of bypassing bad data detection (BDD), while arbitrarily compromising the integrity of state estimators and robust controller even with very sparse measurements corruption. Moreover, based on the inherent sparsity of pragmatic attack signals, ℓ1 -minimization scheme has been used extensively to improve the design of attack-resilient estimators. For this, the theoretical maximum for the percentage of compromised nodes that can be accommodated has been shown to be 50%. In order to guarantee correct state recoveries for larger percentage of attacked nodes, researchers have begun to incorporate prior information into the underlying resilient observer design framework. For the most pragmatic cases, this prior information is often obtained through some data-driven machine learning process. Existing results have shown strong positive correlation between the tolerated attack percentages and the precision of the prior information. In this paper, we present a pruning method to improve the precision of the prior information, given corresponding stochastic uncertainty characteristics of the underlying machine learning model. Then a weighted ℓ1 -minimization is proposed based on the pruned prior. The theoretical and simulation results show that the pruning method significantly improves the observer performance for much larger attack percentages, even when moderately accurate machine learning model used.

Resilient observer, Cyber-physical systems, prunin↗
NASA NTRS1993

Extending model-based diagnosis for analog thermodynamical devices

The increasing complexity of process control applications have posed difficult problems in fault detection, isolation, and recovery. Deep knowledge-based approaches, such as model-based diagnosis, have offered some promise in addressing these problems. However, the difficulties of adapting these techniques to situations involving numerical reasoning and noise have limited the applicability of these techniques. This paper describes an extension of classical model-based diagnosis techniques to deal with sparse data, noise, and complex noninvertible numerical models. These diagnosis techniques are being applied to the External Active Thermal Control System for Space Station Freedom.

Rouquette, Nicolas↗
NASA NTRS1990

Australian topography from Seasat overland altimetry

Retracking of overland returns from the Seasat altimeter using algorithms originally developed for recovering elevations over ice has led to the successful recovery of high quality continental topography over Australia and other continents. Cross-over analysis both before and after orbit adjustment shows the altimetric data over land to have a 2-3 m quality. Direct comparison of gridded Seasat data with surface data re-averaged in the same way shows excellent agreement except where Seasat data are sparse, due either to poor track spacing or to dropouts caused by loss of tracker lock over steeply sloping ground. These results suggest that useful topographic data can be derived from Seasat and the more recent Geosat altimeters for parts of the world where surface data are few or of poor quality.

Frey, Herbert↗
DOE OSTI2021

Global field reconstruction from sparse sensors with Veronoi tessellation-assisted deep learning

Achieving accurate and robust global situational awareness of a complex time-evolving field from a limited number of sensors has been a longstanding challenge. This reconstruction problem is especially difficult when sensors are sparsely positioned in a seemingly random or unorganized manner, which is often encountered in a range of scientific and engineering problems. Moreover, these sensors can be in motion and can become online or offline over time. The key leverage in addressing this scientific issue is the wealth of data accumulated from the sensors. As a solution to this problem, we propose a data-driven spatial field recovery technique founded on a structured grid-based deep-learning approach for arbitrary positioned sensors of any numbers. It should be noted that the naïve use of machine learning becomes prohibitively expensive for global field reconstruction and is furthermore not adaptable to an arbitrary number of sensors. In the present work, we consider the use of Voronoi tessellation to obtain a structured-grid representation from sensor locations enabling the computationally tractable use of convolutional neural networks. One of the central features of the present method is its compatibility with deep-learning based super-resolution reconstruction techniques for structured sensor data that are established for image processing. The proposed reconstruction technique is demonstrated for unsteady wake flow, geophysical data, and three-dimensional turbulence. The current framework is able to handle an arbitrary number of moving sensors, and thereby overcomes a major limitation with existing reconstruction methods. The presented technique opens a new pathway towards the practical use of neural networks for real-time global field estimation.

Fukami, Kai↗
NASA NTRS1988

Thermally reversible gels in electrophoresis. I - Matrix characterization

Two series of thermally reversible hydrogen-bonded gels have been characterized: (5 pct) PVA-(4 pct) PEG and (5 pct) PVA-(0.04 pct) borate gels. They both have extremely low melting points (16-17 C) and could be of potential interest for recovery of proteins after preparative electrophoresis. The PVA-borate gels can be exploited in the pH range 7-11 by progressively increasing the borate content in the pH interval 8 to 7 and concomitantly decreasing the borate levels in the pH zone 8 to 11. It is hypothesized that the low melting point of these gels is due to the fact that they are sparingly and sparsely hydrogen bonded along the PVA chain: on the average, 1 OH group out of 3 or 4 OH groups in the PVA polymer should be engaged in H-bond formation.

Righetti, Pier Giorgio↗
NASA NTRS2006

Thermal Band Atmospheric Correction Using Atmospheric Profiles Derived from Global Positioning System Radio Occultation and the Atmospheric Infrared Sounder

This Rapid Prototyping Capability study explores the potential to use atmospheric profiles derived from GPS (Global Positioning System) radio occultation measurements and by AIRS (Atmospheric Infrared Sounder) onboard the Aqua satellite to improve surface temperature retrieval from remotely sensed thermal imagery. This study demonstrates an example of a cross-cutting decision support technology whereby NASA data or models are shown to improve a wide number of observation systems or models. The ability to use one data source to improve others will be critical to the GEOSS (Global Earth Observation System of Systems) where a large number of potentially useful systems will require auxiliary datasets as input for decision support. Atmospheric correction of thermal imagery decouples TOA radiance and separates surface emission from atmospheric emission and absorption. Surface temperature can then be estimated from the surface emission with knowledge of its emissivity. Traditionally, radiosonde sounders or atmospheric models based on radiosonde sounders, such as the NOAA (National Oceanic & Atmospheric Administration) ARL (Air Resources Laboratory) READY (Real-time Environmental Application and Display sYstem), provide the atmospheric profiles required to perform atmospheric correction. Unfortunately, these types of data are too spatially sparse and too infrequently taken. The advent of high accuracy, global coverage, atmospheric data using GPS radio occultation and AIRS may provide a new avenue for filling data input gaps. In this study, AIRS and GPS radio occultation derived atmospheric profiles from the German Aerospace Center CHAMP (CHAllenging Minisatellite Payload), the Argentinean Commission on Space Activities SAC-C (Satellite de Aplicaciones Cientificas-C), and the pair of NASA GRACE (Gravity Recovery and Climate Experiment) satellites are used as input data in atmospheric radiative transport modeling based on the MODTRAN (MODerate resolution atmospheric TRANsmittance) radiative transport software to separate out the atmospheric component of measured top of atmosphere radiance. Simulated water bodies across a variety of MODTRAN model atmospheres including desert, mid-latitude, tropical and sub-artic conditions provide test bed conditions. Atmospherically corrected radiance and surface temperature results were compared to those obtained using traditional radiosonde balloon data and models. In general, differences between the different techniques were less than 2 percent indicating the potential value satellite derived atmospheric profiles have to atmospherically correct thermal imagery.

Pagnutti, Mary↗
DOE OSTI2020

CO 2 Injection Monitoring with a Scalable, Automated Sparse Seismic Array

Conference presentation at American Institute of Chemical Engineers (AIChE) Annual Meeting, Virtual, November 16–20, 2020. The scalable, automated sparse seismic array (SASSA) method has been ideated as a cost-effective alternative or complementary to conventional 3D seismic methods. Based on a sparse array of receivers deployed strategically on Earth’s surface and one or more fixed seismic sources, the SASSA method can be used to study specific portions of the subsurface to detect the movement of CO 2 in a reservoir with lower environmental footprint and shorter turnaround time to deliver results than conventional 3D seismic methods. Energy & Environmental Research Center (EERC) carried out seismic monitoring of an oilfield undergoing CO 2 enhanced oil recovery in Montana, USA, using the SASSA method.

02 PETROLEUM↗
DOE OSTI2021

Sulfur Poisoning and Performance Recovery of SOFC Air Electrodes

The sulfur poisoning and performance recovery of the state-of-the-art SOFC cathodes (La 0 . 80 Sr 0 . 20 ) 0 . 95 MnO 3 ± δ (LSM) and (La 0 . 60 Sr 0 . 40 ) 0 . 95 Co 0 . 20 Fe 0 . 80 O 3 – δ (LSCF), have been studied. Electrochemical impedance spectroscopy measurements of LSCF|GDC and LSM|YSZ half-cells are carried out in alternating atmospheres of air and SO 2 –air at 700°C for hundreds of hours. In the presence of SO 2 , the electrochemical performance of both the cells decays with ohmic and non-ohmic losses, owing to the absorption and chemical interaction of SO 2 with the electrodes. In LSCF, the SrO segregated on the surface tends to absorb and react with SO 2 , forming SrSO 4 followed by the exsolution of Co-Fe. As for LSM, SO 2 is absorbed onto the Sr-rich areas of LSM, including the active reaction sites near the TPBs, leading to Sr exsolution and SrSO 4 formation, leaving a Sr-deficient LSM. During the subsequent exposure to air, the performance of the sulfur-contaminated LSM is almost restored. The LSM particles, exposed to alternating atmospheres of air and SO 2 -air during the electrochemical tests, show a relatively clean surface with sparsely distributed SrSO 4 particles, indicating a high stability against sulfur poisoning. It is suggested that the loosely adsorbed SO 2 at the TPBs is readily swept away by the SO 2 -free air flow, recovering its ORR activity, whereas the Sr-deficient LSM due to Sr-exsolution stays modified, contributing to the incomplete performance restoration. Unlike the case of LSM, the performance of the sulfur-poisoned LSCF partially recovers during the subsequent exposure to air. Correspondingly, the LSCF particles have a modified morphology covered with numerous nanoparticles, mostly SrSO 4 , showing the irreversible aspect of the sulfur poisoning. The morphology modification is not concentrated near the electrode/electrolyte interface but over the entire cathode, indicating that the degree of recovery from sulfur poisoning is closely related to the presence of SrO and chemical activity of Sr in the electrodes at the solid-gas interface. These results also show the potential application of LSM for a sulfur sensor available in high-temperature harsh conditions.

25 ENERGY STORAGE↗
DOE OSTI2025

Nature-inspired lotus-shaped fins combined with hybrid nanoparticles and metal foam for high-performance latent heat thermal energy storage

Latent heat thermal energy storage (LHTES) systems play a critical role in renewable energy integration by providing high energy density and nearly isothermal operation during phase transitions. However, their performance is often limited by slow melting/charging rates, which motivates the search for enhanced heat transfer designs. This study investigates the melting behavior of RT-82 phase change material (PCM) using novel lotus-shaped fins combined with copper metal foam and conductive graphene nanoparticles and carbon nanotubes. A two-dimensional enthalpy-porosity model in ANSYS Fluent was developed to simulate the charging/melting process, capturing non-thermal equilibrium between the foam and PCM/nano-PCM. In this study, effects of fin geometry, nanoparticle concentration, and foam porosity on melting dynamics and cost-performance trade-offs were investigated. Results showed that natural convection accelerated melting by ~12% compared to conduction-only scenarios. Optimized lotus-shaped fins with higher fin density (T3F4 and T3F10) achieved up to 63% faster melting relative to sparse configurations. Graphene nanoparticles improved thermal conductivity, with a 6% volume fraction, by reducing melting time by ~6.9%, while their combination with 75% porosity foam achieved a maximum reduction in the melting time of ~51% compared to pure PCM. Cost-performance analysis identified T3F4 as the most balanced design, offering rapid thermal response without excessive material costs, while moderate-density designs like T3S6 provided economical alternatives with acceptable performance. These results highlight the performance enhancement that can be achieved by integrating bio-inspired fins, nanoparticles, and foams, into compact and efficient LHTES for solar heating, building thermal management, and industrial waste-heat recovery applications.

25 ENERGY STORAGE↗
DOE OSTI2022

Iterative reconstruction excursions for Baryon Acoustic Oscillations and beyond

ABSTRACT The density field reconstruction technique has been widely used for recovering the baryon acoustic oscillation (BAO) feature in galaxy surveys that has been degraded due to non-linearities. Recent studies advocated adopting iterative steps to improve the recovery much beyond that of the standard technique. In this paper, we investigate the performance of a few selected iterative reconstruction techniques focusing on the BAO and the broad-band shape of the two-point clustering. We include redshift-space distortions, halo bias, and shot noise and inspect the components of the reconstructed field in Fourier space and in configuration space using both density field-based reconstruction and displacement field-based reconstruction. We find that the displacement field reconstruction becomes quickly challenging in the presence of non-negligible shot noise and therefore present surrogate methods that can be practically applied to a much more sparse field such as galaxies. For a galaxy field, implementing a debiasing step to remove the Lagrangian bias appears crucial for the displacement field reconstruction. We show that the iterative reconstruction does not substantially improve the BAO feature beyond an aggressively optimized standard reconstruction with a small smoothing kernel. However, we find taking iterative steps allows us to use a small smoothing kernel more ‘stably’, i.e. without causing a substantial deviation from the linear power spectrum on large scales. In one specific example we studied, we find that a deviation of 13 per cent in $P(k\sim 0.1\, h{\rm \,\,Mpc^{-1}})$ with an aggressive standard reconstruction can reduce to 3–4 per cent with iterative steps.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2011

Estimating Evapotranspiration Using an Observation Based Terrestrial Water Budget

Evapotranspiration (ET) is difficult to measure at the scales of climate models and climate variability. While satellite retrieval algorithms do exist, their accuracy is limited by the sparseness of in situ observations available for calibration and validation, which themselves may be unrepresentative of 500m and larger scale satellite footprints and grid pixels. Here, we use a combination of satellite and ground-based observations to close the water budgets of seven continental scale river basins (Mackenzie, Fraser, Nelson, Mississippi, Tocantins, Danube, and Ubangi), estimating mean ET as a residual. For any river basin, ET must equal total precipitation minus net runoff minus the change in total terrestrial water storage (TWS), in order for mass to be conserved. We make use of precipitation from two global observation-based products, archived runoff data, and TWS changes from the Gravity Recovery and Climate Experiment satellite mission. We demonstrate that while uncertainty in the water budget-based estimates of monthly ET is often too large for those estimates to be useful, the uncertainty in the mean annual cycle is small enough that it is practical for evaluating other ET products. Here, we evaluate five land surface model simulations, two operational atmospheric analyses, and a recent global reanalysis product based on our results. An important outcome is that the water budget-based ET time series in two tropical river basins, one in Brazil and the other in central Africa, exhibit a weak annual cycle, which may help to resolve debate about the strength of the annual cycle of ET in such regions and how ET is constrained throughout the year. The methods described will be useful for water and energy budget studies, weather and climate model assessments, and satellite-based ET retrieval optimization.

Rodell, Matthew↗
DOE OSTI2022

Comprehensive compartmental model and calibration algorithm for the study of clinical implications of the population-level spread of COVID-19: a study protocol

The complex dynamics of the coronavirus disease 2019 (COVID-19) pandemic has made obtaining reliable long-term forecasts of the disease progression difficult. Simple mechanistic models with deterministic parameters are useful for short-term predictions but have ultimately been unsuccessful in extrapolating the trajectory of the pandemic because of unmodelled dynamics and the unrealistic level of certainty that is assumed in the predictions. We propose a 22-compartment epidemiological model that includes compartments not previously considered concurrently, to account for the effects of vaccination, asymptomatic individuals, inadequate access to hospital care, post-acute COVID-19 and recovery with long-term health complications. Additionally, new connections between compartments introduce new dynamics to the system and provide a framework to study the sensitivity of model outputs to several concurrent effects, including temporary immunity, vaccination rate and vaccine effectiveness. Subject to data availability for a given region, we discuss a means by which population demographics (age, comorbidity, socioeconomic status, sex and geographical location) and clinically relevant information (different variants, different vaccines) can be incorporated within the 22-compartment framework. Considering a probabilistic interpretation of the parameters allows the model's predictions to reflect the current state of uncertainty about the model parameters and model states. We propose the use of a sparse Bayesian learning algorithm for parameter calibration and model selection. This methodology considers a combination of prescribed parameter prior distributions for parameters that are known to be essential to the modelled dynamics and automatic relevance determination priors for parameters whose relevance is questionable. This is useful as it helps prevent overfitting the available epidemiological data when calibrating the parameters of the proposed model. Population-level administrative health data will serve as partial observations of the model states.

59 BASIC BIOLOGICAL SCIENCES↗
NASA NTRS2012

Application of Satellite Gravimetry for Water Resource Vulnerability Assessment

The force of Earth's gravity field varies in proportion to the amount of mass near the surface. Spatial and temporal variations in the gravity field can be measured via their effects on the orbits of satellites. The Gravity Recovery and Climate Experiment (GRACE) is the first satellite mission dedicated to monitoring temporal variations in the gravity field. The monthly gravity anomaly maps that have been delivered by GRACE since 2002 are being used to infer changes in terrestrial water storage (the sum of groundwater, soil moisture, surface waters, and snow and ice), which are the primary source of gravity variability on monthly to decadal timescales after atmospheric and oceanic circulation effects have been removed. Other remote sensing techniques are unable to detect water below the first few centimeters of the land surface. Conventional ground based techniques can be used to monitor terrestrial water storage, but groundwater, soil moisture, and snow observation networks are sparse in most of the world, and the countries that do collect such data rarely are willing to share them. Thus GRACE is unique in its ability to provide global data on variations in the availability of fresh water, which is both vital to life on land and vulnerable to climate variability and mismanagement. This chapter describes the unique and challenging aspects of GRACE terrestrial water storage data, examples of how the data have been used for research and applications related to fresh water vulnerability and change, and prospects for continued contributions of satellite gravimetry to water resources science and policy.

Rodell, Matthew↗
NASA NTRS2014

Reevaluation of Stratospheric Ozone Trends From SAGE II Data Using a Simultaneous Temporal and Spatial Analysis

This paper details a new method of regression for sparsely sampled data sets for use with time-series analysis, in particular the Stratospheric Aerosol and Gas Experiment (SAGE) II ozone data set. Non-uniform spatial, temporal, and diurnal sampling present in the data set result in biased values for the long-term trend if not accounted for. This new method is performed close to the native resolution of measurements and is a simultaneous temporal and spatial analysis that accounts for potential diurnal ozone variation. Results show biases, introduced by the way data is prepared for use with traditional methods, can be as high as 10%. Derived long-term changes show declines in ozone similar to other studies but very different trends in the presumed recovery period, with differences up to 2% per decade. The regression model allows for a variable turnaround time and reveals a hemispheric asymmetry in derived trends in the middle to upper stratosphere. Similar methodology is also applied to SAGE II aerosol optical depth data to create a new volcanic proxy that covers the SAGE II mission period. Ultimately this technique may be extensible towards the inclusion of multiple data sets without the need for homogenization.

Damadeo, R. P.↗
DOE OSTI2022

Dynamic signal recovery in distribution grids using compressive lossy measurements

Distribution system state estimation requires reliable aggregation of the measured data. However, the large volume of the measured data imposes a significant stress on the underlying communication infrastructure. With the challenges associated with measurement availability, current distribution systems are typically unobservable. To cope with the unobservability issue, compressive sensing theory allows us to recover system state information from a small number of measurements provided the states of the distribution system exhibit sparsity. In this paper, we evaluate the robustness of an updated Kalman filtered modified compressive sensing (KF-ModCS) technique that dynamically estimates the grid states using a small fraction of measured data. In practice, measurements used for sparsity based state estimation may also be intermittent due to communication network induced losses. Further, to understand the effect of packet losses on KF-ModCS, we provide an upper bound for the expected variances of the state estimation error for a given rate of information loss. This upper bound is further improved if the support set of the sparse signal that characterizes the state dynamics does not change over time and/or the reduced model is observable. Simulations based on two practical data sets collected from actual customers in a distribution grid validate the theoretical results.

24 POWER TRANSMISSION AND DISTRIBUTION↗