Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Distributed Computing Resources”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 685 records · Page 38
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

EMT-TS Hybrid Simulation for Large Power Grids Considering IBR-Driven Dynamics

The escalating integration of inverter-based resources (IBRs) poses new challenges to power systems by introducing fast dynamics with higher frequencies, which may need to be simulated by an electromagnetic transient (EMT) program. As an alternative to conducting EMT simulations for the entire system, which is typically time consuming, hybrid simulation between EMT and phasor-domain transient stability (TS) can greatly reduce the computational burden while preserving the detailed fast dynamics in the EMT zone. This paper establishes an EMT-TS hybrid simulation platform using open-source tools, specifically ParaEMT, GridPACK, and HELICS, which are the EMT simulator, TS simulator, and interface framework, respectively. Case studies on the 240-bus Western Electricity Coordinating Council (WECC) system demonstrate that the developed ParaEMT-HELICS-GridPACK hybrid simulator can accurately capture both slow electromechanical and fast IBR-driven dynamics with a 2.4x speedup.

electromagnetic transient↗
DOE OSTI2021

Communication-Avoiding and Memory-Constrained Sparse Matrix-Matrix Multiplication at Extreme Scale

Sparse matrix-matrix multiplication (SpGEMM) is a widely used kernel in various graph, scientific computing and machine learning algorithms. In this paper, we consider SpGEMMs performed on hundreds of thousands of processors generating trillions of nonzeros in the output matrix. Distributed SpGEMM at this extreme scale faces two key challenges: (1) high communication cost and (2) inadequate memory to generate the output. Furthermore, we address these challenges with an integrated communication-avoiding and memory-constrained SpGEMM algorithm that scales to 262,144 cores (more than 1 million hardware threads) and can multiply sparse matrices of any size as long as inputs and a fraction of output fit in the aggregated memory. As we go from 16,384 cores to 262,144 cores on a Cray XC40 supercomputer, the new SpGEMM algorithm runs 10x faster when multiplying large-scale protein-similarity matrices.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Performance Testing and Assessment of Protection Scheme Using Real-Time Hardware-in-the-Loop and IEC 61850 Standard

The main challenge of the microgrid is to design a suitable protection scheme due to the complexity of the architecture of the microgrid. The importance of the proposed protection technique is threefold. First, it presents a co-simulation platform to integrate between a simulated model on power system computer aided design (PSCAD)/real time digital simulator computer aided design (RSCAD) software’s and physical devices schweitzer engineering laboratories (SEL) 421-7 relays to protect the microgrid that includes different resources connected based on inverter interface. Second, it presents a comprehensive hardware/software setup to test the protective relays in a closed loop system and shows how to configure the protective relay’s International Electrotechnical Commission 61850 communications. Third, IEEE 1588 standard is used to provide sub nanoseconds latency between the simulated model that emulated on real time digital simulator (RTDS) and the external devices. Also, the measurement signals are synchronized between RTDS and the external devices using giga-transceiver synchronization card (GTSYNC) interface card and SEL-2488 satellite-synchronized network clock. The results showed that the co-simulation infrastructure introduces a highly dependable design, analysis, and testing environment for cyber and physical data flow in the system. Besides that, the voltages at ac/dc sides and frequency at fault condition were maintained due to the energy storage device contributions at different modes of operation.

42 ENGINEERING↗
DOE OSTI2023

Simulations of Fuel-Air Mixing in a 7 Element Lean Direct Injection (LDI) Aviation Combustor: Preprint

The increased accessibility of commercial aviation to the general population has in-creased the conventional jet fuel consumption, thereby causing increased CO2emissions. One of the strategies to reduce the environmental impact is to use sustainable aviation fuels (SAFs). SAFs are jet fuels obtained from renewable resources such as biomass and waste streams such as plastics/municipal solid wastes. Even though the use of SAFs has been gaining importance over the last two decades, only a few numerical studies on their usage have been reported so far. In this study, two SAFs namely, alcohol to jet (ATJ) and hydrotreated esters and fatty acids (HEFA), are chosen and their use in a simple, lead direct injection (LDI) combustor is studied. The focus of this study is limited to the thermo-chemical properties of the fuels and their impact on fuel distribution and evaporation. Hence, only non-reactive simulations are performed. A high accuracy, adaptive mesh refinement based low-Mach solver, PeleLMeX, is used for performing the simulations. The time-averaged flow results indicate faster evaporation of HEFA when compared to ATJ and Jet-A. The difference in fuel evaporation rates is attributed to the differences in fuel viscosity and surface tension properties and the concomitant variations in the fuel droplet size distributions.

adaptive mesh refinement↗
DOE OSTI2023

GPU-resident sparse direct linear solvers for alternating current optimal power flow analysis

Integrating renewable resources within the transmission grid at a wide scale poses significant challenges for economic dispatch as it requires analysis with more optimization parameters, constraints, and sources of uncertainty. This motivates the investigation of more efficient computational methods, especially those for solving the underlying linear systems, which typically take more than half of the overall computation time. In this paper, we present our work on sparse linear solvers that take advantage of hardware accelerators, such as graphical processing units (GPUs), and improve the overall performance when used within economic dispatch computations. We treat the problems as sparse, which allows for faster execution but also makes the implementation of numerical methods more challenging. We present the first GPU-native sparse direct solver that can execute on both AMD and NVIDIA GPUs. We demonstrate significant performance improvements when using high-performance linear solvers within alternating current optimal power flow (ACOPF) analysis. Furthermore, we demonstrate the feasibility of getting significant performance improvements by executing the entire computation on GPU-based hardware. Finally, we identify outstanding research issues and opportunities for even better utilization of heterogeneous systems, including those equipped with GPUs.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2023

Assessing the Optimality of LinDist3Flow for Optimal Tap Selection of Step Voltage Regulators in Unbalanced Distribution Networks

The adoption of distributed energy resources such as photovoltaics (PVs) has increased dramatically during the previous decade. The increased penetration of PVs into distribution networks (DNs) can cause voltage fluctuations that have to be mitigated. One of the key utility assets employed to this end are step-voltage regulators (SVRs). It is desirable to include tap selection of SVRs in optimal power flow (OPF) routines, a task that turns out to be challenging because the resultant OPF problem is nonconvex with added complexities stemming from accurate SVR modeling. While several convex relaxations based on semi-definite programming (SDP) have been presented in the literature for optimal tap selection, SDP based schemes do not scale well and are challenging to implement in large-scale planning or operational frameworks. This paper deals with the optimal tap selection (OPTS) problem for wye-connected SVRs using linear approximations of power flow equations. Specifically, the LinDist3Flow model is adopted and the effective SVR ratio is assumed to be continuous- enabling the formulation of a problem called LinDist3Flow-OPTS, which amounts to a linear program. The scalability and optimality gap of LinDist3Flow-OPTS are evaluated with respect to existing SDP-based and nonlinear programming techniques for optimal tap selection in three standard feeders, namely, the IEEE 13-bus, 123-bus, and 8500-node DNs. For all DNs considered, LinDist3Flow-OPTS achieves an optimality gap of approximately 1% or less while significantly lowering the computational burden.

linear approximations↗
DOE OSTI2022

Assessing the Optimality of LinDist3Flow for Optimal Tap Selection of Step Voltage Regulators in Unbalanced Distribution Networks: Preprint

The adoption of distributed energy resources such as photovoltaics (PVs) has increased dramatically during the previous decade. The increased penetration of PVs into distribution networks (DNs) can cause voltage fluctuations that have to be mitigated. One of the key utility assets employed to this end are step-voltage regulators (SVRs). It is desirable to include tap selection of SVRs in optimal power flow (OPF) routines, a task that turns out to be challenging because the resultant OPF problem is nonconvex with added complexities stemming from accurate SVR modeling. While several convex relaxations based on semi-definite programming (SDP) have been presented in the literature for optimal tap selection, SDP based schemes do not scale well and are challenging to implement in large-scale planning or operational frameworks. This paper deals with the optimal tap selection (OPTS) problem for wye-connected SVRs using linear approximations of power flow equations. Specifically, the LinDist3Flow model is adopted and the effective SVR ratio is assumed to be continuous–enabling the formulation of a problem called LinDist3Flow-OPTS, which amounts to a linear program. The scalability and optimality gap of LinDist3Flow-OPTS are evaluated with respect to existing SDP-based and nonlinear programming techniques for optimal tap selection in three standard feeders, namely, the IEEE 13-bus, 123-bus, and 8500-node DNs. For all DNs considered, LinDist3Flow-OPTS achieves an optimality gap of approximately 1% or less while significantly lowering the computational burden.

linear approximations↗
DOE OSTI2026

Framework for Modeling 3D-Printed Concrete Construction to Assess Energy Efficiency and Backup Power Trade-Offs in a Mixed-Use, New Construction Neighborhood Development

This paper presents a framework that expands the URBANopt(TM) modeling platform to include 3D-printed concrete wall assemblies and assess energy efficiency and backup power trade-offs in new housing developments. Applied to a planned mixed-use neighborhood in Oil City, Pennsylvania, the workflow integrates building energy and distributed energy resource (DER) modeling to evaluate envelope and equipment upgrades alongside DER operations. Results show that advanced 3D-printed envelopes combined with efficient systems and onsite photovoltaics (PV) and storage reduce energy use intensity and sustain critical loads during outages. The framework supports planning for emerging construction technologies by quantifying key trade-offs between energy efficiency and backup power performance.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2023

Robust Real-Time Modeling of Distribution Systems with Data-Driven Grid-Wise Observability (Final Technical Report)

The overall objective of this project is to leverage existing and emerging sensor measurements to develop data-driven observability enhancement algorithms as well as robust state estimation and parameter identification techniques to enable real-time grid-wise monitoring and modeling of loads and distributed energy resources (DERs). The project resulted in a holistic framework with the following key components: 1) data-driven and machine learning-based grid-edge visibility enhancement, 2) robust branch-current-based state estimation (BCSE), and 3) robust real-time steady-state and dynamic-state modeling of loads and DERs. The research outcomes have provided utility companies better network visibility, higher-fidelity load/DER models, and more accurate assessment of DERs’ impacts, thus, facilitating the integration of renewable energy sources. The synergistic collaborative project among Iowa State University (ISU), Argonne National Laboratory (ANL), Electric Power Research Center (EPRC), SIEMENS Industry, Alliant Energy, Cedar Falls Utilities (CFU), and Maquoketa Valley Electric Cooperative (MVEC) to leverage the team’s extensive expertise and experience in power distribution systems, state estimation, online modeling and identification, and DER integrations. The proposed frameworks have been verified with industry adopted software and attempted integrated with existing tool wherever possible

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2021

Discovering Hidden Geothermal Signatures using Unsupervised Machine Learning

Discovering hidden geothermal resources is a very challenging task. It requires the mining of large datasets, including various diverse data attributes representing subsurface hydrogeological and geothermal conditions. The commonly used Play Fairway Analysis (PFA) typically relies on subject-matter expertise to analyze site or regional data to estimate geothermal conditions and prospectivity. Here, we demonstrate an alternative approach based on machine learning (ML) to process a geothermal dataset of Southwest New Mexico (SWNM). The study region includes low- and medium-temperature hydrothermal systems. However, most of these systems are poorly characterized because of insufficient existing data and limited past explorative studies. This study aims to discover hidden patterns and relationships in the SWNM geothermal dataset to better understand regional hydrothermal conditions. This is achieved by applying an unsupervised machine learning algorithm based on non-negative matrix factorization coupled with customized k-means clustering (NMFk). NMFk can automatically identify (1) hidden (latent) signatures characterizing datasets, (2) the optimal number of these signatures, (3) dominant data attributes associated with each signature, and (4) spatial distribution of the extracted signatures. Here, NMFk is applied to analyze 18 geological, geophysical, hydrogeological, geothermal attributes at 44 locations in SWNM. NMFk successfully finds data patterns and identifies the spatial associations of hydrothermal signatures with the four physiographic provinces in SWNM (Colorado Plateau, Volcanic Field, Basin and Range, and the Rio Grande rift). The algorithm identified up to 5 hydrothermal signatures in the SWNM datasets that differentiate between low- and medium-temperature hydrothermal systems in different provinces. Also, the algorithm identifies two medium-temperature hydrothermal systems in SWNM that require further exploration for geothermal resource development. Based on our analyses, 12 of the attributes are important to identify medium-temperature hydrothermal systems, and the remaining six attributes are critical to characterize low-temperature hydrothermal systems. Based on the obtained results, we identify potential physiographic provinces for further exploration to characterize them as geothermal resources. The resulting NMFk model can be applied to predict geothermal conditions and their uncertainties at new SWNM locations based on limited data from unexplored areas.

58 GEOSCIENCES↗
DOE OSTI2022

A computationally efficient algorithm for computing convex hull prices

Electricity markets worldwide allow participants to bid non-convex production offers. While non-convex offers can more accurately reflect a resource's capabilities, they create challenges for market clearing processes. For example, system operators may be required to execute side payments to participants whose costs are not covered through energy sales as determined via traditional locational marginal pricing schemes. Convex hull pricing minimizes this and other types of side payments while providing uniform (i.e., locationally and temporally consistent) prices. Computing convex hull prices involves solving either a large-scale linear program or the Lagrangian dual of the corresponding non-convex scheduling problem. Further, the former approach requires explicit descriptions of market participants' convex hulls. While linear programs for computing convex hull prices are large, their structure is naturally decomposable by generators. Here, in this work, we propose and empirically analyze a Benders decomposition approach to computing convex hull prices that leverages recent advances in convex hull formulations for thermal generating units. We demonstrate across a large set of test instances that our decomposition approach only requires modest computational effort, obtaining solutions at least an order of magnitude faster than the equivalent large-scale linear programming approach. Overall, we provide a computationally feasible method for computing convex hull prices for industrial scale market clearing problems, enabling the possibility of practical adoption of this advanced pricing mechanism.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2021

Hydrological analysis in watersheds with a variable-resolution global climate model (VR-CESM)

Traditionally, watershed-scale hydrology is simulated by distributed hydrological models with offline meteorological forcing data, or by regional regional climate models that link atmospheric and land hydrology interactions. Global climate models (GCMs) are rarely used to study watershed-scale hydrology due to the relatively coarse grid resolution, computationally expensive downscaling, and simplified physical processes. Recently, however, watershed-scale hydrology analysis has become possible in GCMs due to the development of variable-resolution GCMs that dynamically couple the hydrological processes between atmosphere and land systems at fine resolutions in selected regions and coarse resolution elsewhere. In this study, we used the variable-resolution Community Earth System Model (VR-CESM) with refined-resolution (14 km) in the western U.S. and eastern China to evaluate watershed-scale hydrology. Furthermore, we compared the historical VR-CESM outputs with gauge measurements and other hydrological models (e.g., National Water Model in the U.S.) and calibrated the subsurface runoff capacities in four mountainous watersheds. An RCP8.5 projection from 2007 to 2050 is used to estimate the impact of changing precipitation and snow climatology on watershed hydrology. We also analyzed the long-term runoff variability and the possibility of extreme runoff events as simulated by the VR-CESM. Although calibration is not possible in larger-scale watersheds, VR-CESM simulates the long-term annual variability of watersheds and provides insights on climate change impacts on hydrology. We conclude that refined-resolution VR-CESM can be used for watershed-scale hydrology analysis to understand water resources and water balance, in addition to traditional watershed-scale hydrological models. It enables hydrological analysis at multiple watersheds in one simulation and can help understand the two-way dynamics between land surface hydrology and atmospheric processes, and is especially practical for projecting climate change impacts. However, it is challenging to apply VR-CESM for hydrologic analysis in regulated watersheds as human factors (e.g., pumping, irrigation, water diversion) have not been fully addressed in VR-CESM.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Chemical-damage MINE: A database of curated and predicted spontaneous metabolic reactions

Spontaneous reactions between metabolites are often neglected in favor of emphasizing enzyme-catalyzed chemistry because spontaneous reaction rates are assumed to be insignificant under physiological conditions. However, synthetic biology and engineering efforts can raise natural metabolites' levels or introduce unnatural ones, so that previously innocuous or nonexistent spontaneous reactions become an issue. Problems arise when spontaneous reaction rates exceed the capacity of a platform organism to dispose of toxic or chemically active reaction products. While various reliable sources list competing or toxic enzymatic pathways' side-reactions, no corresponding compilation of spontaneous side-reactions exists, nor is it possible to predict their occurrence. We addressed this deficiency by creating the Chemical Damage (CD)-MINE resource. First, we used literature data to construct a comprehensive database of metabolite reactions that occur spontaneously in physiological conditions. We then leveraged this data to construct 148 reaction rules describing the known spontaneous chemistry in a substrate-generic way. We applied these rules to all compounds in the ModelSEED database, predicting 180,891 spontaneous reactions. The resulting (CD)-MINE is available at https://minedatabase.mcs.anl. gov/cdmine/#/home and through developer tools. We also demonstrate how damage-prone intermediates and end products are widely distributed among metabolic pathways, and how predicting spontaneous chemical damage helps rationalize toxicity and carbon loss using examples from published pathways to commercial products. We explain how analyzing damage-prone areas in metabolism helps design effective engineering strategies. Finally, we use the CD-MINE toolset to predict the formation of the novel damage product N-carbamoyl proline, and present mass spectrometric evidence for its presence in Escherichia coli.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2020

Policy-Driven Sustainable Saline Drainage Disposal and Forage Production in the Western San Joaquin Valley of California

Environmental policies to address water quality impairments in the San Joaquin River of California have focused on the reduction of salinity and selenium-contaminated subsurface agricultural drainage loads from westside sources. On 31 December 2019, all of the agricultural drainage from a 44,000 ha subarea on the western side of the San Joaquin River basin was curtailed. This policy requires the on-site disposal of all of the agricultural drainage water in perpetuity, except during flooding events, when emergency drainage to the River is sanctioned. The reuse of this saline agricultural drainage water to irrigate forage crops, such as ‘Jose’ tall wheatgrass and alfalfa, in a 2428 ha reuse facility provides an economic return on this pollutant disposal option. Irrigation with brackish water requires careful management to prevent salt accumulation in the crop root zone, which can impact forage yields. The objective of this study was to optimize the sustainability of this reuse facility by maximizing the evaporation potential while achieving cost recovery. This was achieved by assessing the spatial and temporal distribution of the root zone salinity in selected fields of ‘Jose’ tall wheatgrass and alfalfa in the drainage reuse facility, some of which have been irrigated with brackish subsurface drainage water for over fifteen years. Electromagnetic soil surveys using an EM-38 instrument were used to measure the spatial variability of the salinity in the soil profile. The tall wheatgrass fields were irrigated with higher salinity water (1.2–9.3 dS m -1 ) compared to the fields of alfalfa (0.5–6.5 dS m -1 ). Correspondingly, the soil salinity in the tall wheatgrass fields was higher (12.5 dS m -1 –19.3 dS m -1 ) compared to the alfalfa fields (8.97 dS mm -1 –14.4 dS mm -1 ) for the years 2016 and 2017. Better leaching of salts was observed in the fields with a subsurface drainage system installed (13–1 and 13–2). The depth-averaged root zone salinity data sets are being used for the calibration of the transient hydro-salinity computer model CSUID-ID (a one-dimensional version of the Colorado State University Irrigation Drainage Model). This user-friendly decision support tool currently provides a useful framework for the data collection needed to make credible, field-scale salinity budgets. In time, it will provide guidance for appropriate leaching requirements and potential blending decisions for sustainable forage production. This paper shows the tie between environmental drainage policy and the role of local governance in the development of sustainable irrigation practices, and how well-directed collaborative field research can guide future resource management.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2021

Renewable energy integration and system operation challenge: control and optimization of millions of devices

The electric power infrastructure, originally designed and built on large-scale power plants, is evolving into a more resilient power generation and delivery system in which millions of smaller units of distributed energy generation resources units will be installed in sub-transmission and distribution networks. In order to control, manage and optimize the future grid, a hierarchical design is presented in this chapter which enables the distributed control on grid edge while inheriting the existing centralized control structure. This layered design of large-scale power system operation and control uses the following principle: reactive power control is treated as a primary control for voltage stability, and the real power control is primarily a grid-level control but can also be a supplementary control for voltage support in the case of insufficient reactive power control capacity. For the purpose of active control and operation at the distribution level, a recursive power network model is derived from nodal injection and branch power flow models. Based on the model, the proposed algorithms of hierarchical control, grid-edge inference and dynamic hosting allowance are developed and presented for multi-level controlled operation. And, a co-simulation architecture of integrated T&D system is presented to validate and demonstrate the feasibility and scalability of proposed algorithms.

Xu, Ying↗
DOE OSTI2023

Discovery of False Data Injection Attacks on Power Grid Frequency Controllers with Reinforcement Learning [Poster]

While inverter-based DER (distributed energy resources) are instrumental to integrating renewable energy into the power grid, they reduce the grid's mechanical inertia, thereby increasing the risk of frequency instabilities. To compensate for frequency instability risks, the grid must also undergo a transformation to include digital technologies that allow for two-way communication between the utility and customers. The current and future state of the power grid allows for building a cleaner energy landscape. However, the grid may also become vulnerable to novel cyber threats. To preemptively protect the power grid against elaborate cyber-attacks, we propose to discover potential threats via reinforcement learning. In this work, the focus is on studying false data injection attacks that target the control logic of frequency controllers. We show that a reinforcement learning agent can successfully discover how to best inject false data into linear droop controllers.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2023

Massively parallel modeling and inversion of electrical resistivity tomography data using PFLOTRAN

Abstract. Electrical resistivity tomography (ERT) is a broadly accepted geophysical method for subsurface investigations. Interpretation of field ERT data usually requires the application of computationally intensive forward modeling and inversion algorithms. For large-scale ERT data, the efficiency of these algorithms depends on the robustness, accuracy, and scalability on high-performance computing resources. In this regard, we present a robust and highly scalable implementation of forward modeling and inversion algorithms for ERT data. The implementation is publicly available and developed within the framework of PFLOTRAN, an open-source, state-of-the-art massively parallel subsurface flow and transport simulation code. The forward modeling is based on a finite-volume discretization of the governing differential equations, and the inversion uses a Gauss–Newton optimization scheme. To evaluate the accuracy of the forward modeling, two examples are first presented by considering layered (1D) and 3D earth conductivity models. The computed numerical results show good agreement with the analytical solutions for the layered earth model and results from a well-established code for the 3D model. Inversion of ERT data, simulated for a 3D model, is then performed to demonstrate the inversion capability by recovering the conductivity of the model. To demonstrate the parallel performance of PFLOTRAN's ERT process model and inversion capabilities, large-scale scalability tests are performed by using up to 131 072 processes on a leadership class supercomputer. These tests are performed for the two most computationally intensive steps of the ERT inversion: forward modeling and Jacobian computation. For the forward modeling, we consider models with up to 122 ×106 degrees of freedom (DOFs) in the resulting system of linear equations and demonstrate that the code exhibits almost linear scalability on up to 10 000 DOFs per process. On the other hand, the code shows superlinear scalability for the Jacobian computation, mainly because all computations are fairly evenly distributed over each process with no parallel communication.

58 GEOSCIENCES↗
DOE OSTI2022

Deep Reinforcement Learning for Distribution System Restoration Using Distributed Energy Resources and Tie-Switches

Distributed energy resources (DERs), such as solar PVs and energy storage, can be used to restore distribution system critical loads after the extreme weather events to increase grid resilience. However, coordinating multiple DERs together with tie-switches for multi-step restoration process under renewable uncertainty is challenging. This paper proposes a deep reinforcement learning to control discrete actions of switching on/off tie switches and DERs for critical load restoration. The restoration problem is first cast into the Markov decision process suitable for DRL. Then, the original soft actor critic (SAC) method for continuous actions has been extended to handle discrete and continuous actions. Numerical comparison results with other stochastic optimization-based approaches on the modified IEEE 33-bus system show that the proposed method can achieve fast critical load restoration in the presence of substation power outage while maintaining system voltage limit throughout the restoration process.

active distribution systems↗