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At least 73 records · Page 4

Multi-Area Distribution System State Estimation Using Decentralized Physics-Aware Neural Networks

The development of active distribution grids requires more accurate and lower computational cost state estimation. In this paper, the authors investigate a decentralized learning-based distribution system state estimation (DSSE) approach for large distribution grids. The proposed approach decomposes the feeder-level DSSE into subarea-level estimation problems that can be solved independently. The proposed method is decentralized pruned physics-aware neural network (D-P2N2). The physical grid topology is used to parsimoniously design the connections between different hidden layers of the D-P2N2. Monte Carlo simulations based on one-year of load consumption data collected from smart meters for a three-phase distribution system power flow are developed to generate the measurement and voltage state data. The IEEE 123-node system is selected as the test network to benchmark the proposed algorithm against the classic weighted least squares and state-of-the-art learning-based DSSE approaches. Numerical results show that the D-P2N2 outperforms the state-of-the-art methods in terms of estimation accuracy and computational efficiency.

24 POWER TRANSMISSION AND DISTRIBUTION↗

On the Solution of ℓ 0 -Constrained Sparse Inverse Covariance Estimation Problems

The sparse inverse covariance matrix is used to model conditional dependencies between variables in a graphical model to fit a multivariate Gaussian distribution. Estimating the matrix from data are well known to be computationally expensive for large-scale problems. Sparsity is employed to handle noise in the data and to promote interpretability of a learning model. Although the use of a convex ℓ 1 regularizer to encourage sparsity is common practice, the combinatorial ℓ 0 penalty often has more favorable statistical properties. In this paper, we directly constrain sparsity by specifying a maximally allowable number of nonzeros, in other words, by imposing an ℓ 0 constraint. Here, we introduce an efficient approximate Newton algorithm using warm starts for solving the nonconvex ℓ 0 -constrained inverse covariance learning problem. Numerical experiments on standard data sets show that the performance of the proposed algorithm is competitive with state-of-the-art methods.

$\ell_0$-Constrained↗

CACTI: Best Estimate Aerosol Size Distribution by airborne measurements

These data were collected during the Cloud, Aerosol, and Complex Terrain Interactions (CACTI; https://www.arm.gov/research/campaigns/amf2018cacti ) field campaign in the Sierras de Córdoba mountain range of north-central Argentina as part of ARM Aerial Facility (AAF) deployment. The ARM Aerial Facility Gulfstream-1 was operated from Las Higueras Airport (IATA: RCU, ICAO: SAOC), Río Cuarto, Córdoba, Argentina, for the Intensive Observation Period (IOP) from Nov. 1 through Dec. 15, 2018. The G-1 aircraft performed 22 research flights over the first ARM Mobile Facility (AMF1) location in the Sierras de Córdoba mountain range to measure atmospheric state and turbulence, cloud water content and droplet size distributions, aerosol precursor gases, aerosol chemical composition and size distributions. The current data set presents Best Estimate Aerosol Size Distribution: a merged aerosol size distribution composed of the data from 5 sensors; three aerosol spectrometers: Scanning Mobility Particle Sizer (SMPS), Ultra-High Sensitivity Aerosol Spectrometer (UHSAS), and Passive Cavity Aerosol Spectrometer (PCASP); and two cloud probes: Cloud Aerosol Spectrometer (CAS) and Fast Cloud Droplet Probe (FCDP). The SMPS data were interpolated to 1 second from “native” time resolution of about 64 second to match all other probes.

54 ENVIRONMENTAL SCIENCES↗

NPFTURBULENCE: Best Estimate Aerosol Size Distribution by airborne measurements

The original data were collected during the field campaign of “Turbulent layers promoting New Particle Formation” experiment (NPFTURBULENCE; https://www.arm.gov/research/campaigns/aaf2024npfturbulence) over the Atmospheric Radiation Measurement (ARM) Southern Great Plains (SGP) Atmospheric Observatory (https://www.arm.gov/capabilities/observatories/sgp ) in north-central Oklahoma. The ARM Aerial Facility ArcticShark uncrewed aerial system (UAS) was based at Blackwell–Tonkawa Municipal Airport (IATA: BWL, ICAO: KBKN, FAA LID: BKN, 36.74475° N, 97.34918° W, 313.9 m MSL), for the field campaign from May 5 through May 29, 2024. The ArcticShark UAS performed 11 flights, including 10 research flights over the Central Facility of the ARM SGP to measure atmospheric state, turbulence, surface IT temperature and imagery, aerosol number concentration and size distribution. The current data set presents Best Estimate Aerosol Size Distribution: a merged aerosol size distribution composed of the data from 2 sensors: miniaturized Scanning Electrical Mobility Sizer (mSEMS) and Portable Optical Particle Spectrometer (POPS). The mSEMS data were interpolated to 1 second from “native” time resolution of about 15 second to match the other probe. The POPS data were converted from equivalent optical size into geometric size using value of aerosol refractive index of 1.477 from the HISCALE field campaign (same geographical area, altitudes, and time of year; http://www.arm.gov/campaigns/aaf2016hiscale ).

54 ENVIRONMENTAL SCIENCES↗

Evaluating Retention Index Score Assumptions to Refine GC–MS Metabolite Identification

As metabolomics grows into a high-throughput and high demand research field, current metrics for the identification of small molecules in gas chromatography–mass spectrometry (GC–MS) still require manual verification. Though steps have been taken to improve scoring metrics by combining spectral similarity (SS) and retention index (RI), the problem persists. A large body of literature has analyzed and refined SS scores, but few studies have explicitly studied improvements to RI scores. Here, in this study, we examined whether uninvestigated assumptions of the RI score are valid and propose ways to improve them. Query RIs were matched to library RI with a generous window of ±35 to avoid unintentional removal of valid compound identifications. Each match was manually verified as a true positive (TP), true negative, or unknown. Metabolites with at least 30 TP identifications were included in downstream analyses, resulting in a total of 87 metabolites from samples of varying complexity and type (e.g., amino acid mixtures, human urine, fungal species, and so on.). Our results showed that the RI score assumptions of normality, consistent variance across metabolites, and a mean error centered at 0 are often violated. We demonstrated through a cross-validation analysis that modifying these underlying assumptions according to empirical metabolite-specific distributions improved the TP and negative rankings. Further, we statistically determined the minimum number of samples required to estimate distributional parameters for scoring metrics. Overall, this work proposes a robust statistical pipeline to reduce the time bottleneck of metabolite identification by improving RI scores and thus minimize the effort to complete manual verification.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

An Online Joint Optimization–Estimation Architecture for Distribution Networks

Here in this article, we propose an optimal joint optimization-estimation architecture for distribution networks, which jointly solves the optimal power flow (OPF) problem and static state estimation (SE) problem through an online gradient-based feedback algorithm. The main objective is to enable a fast and timely interaction between the OPF decisions and state estimators with limited sensor measurements. First, convergence and optimality of the proposed algorithm are analytically established. Then, the proposed gradient-based algorithm is modified by introducing statistical information of the inherent estimation and linearization errors for an improved and robust performance of the online OPF decisions. Overall, the proposed method eliminates the traditional separation of operation and monitoring, where optimization and estimation usually operate at distinct layers and different time scales. Hence, it enables a computationally affordable, efficient, and robust online operational framework for distribution networks under time-varying settings.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Robust Medium-Voltage Distribution System State Estimation using Multi-Source Data

Due to the lack of sufficient online measurements for distribution system observability, pseudo-measurements from short-term load or distributed renewable energy resources (DERs) forecasting are used. However, the accuracy of them is low and thus significantly limits the performance of distribution system state estimation (DSSE). In this paper, a robust DSSE that integrates multi-source measurement data is proposed. Specifically, the historical low-voltage (LV) side smart meters are used to forecast load and DERs injections via the support vector machine (SVM) with optimally tuned parameters. By contrast, the online smart meters at LV side are utilized to derive equivalent power injections at the MV/LV transformers, yielding more accurate pseudo-measurements compared to the forecasted injections. Furthermore, to deal with bad data caused by communication loss, instrumental errors and cyber attacks, robust DSSE that relies on generalized maximum-likelihood (GM)-estimation criterion is developed. The projection statistics are developed to adjust the weights of each measurement, leading to better balance between pseudo- and real-time measurements. Numerical results conducted on modified IEEE 33-bus system with DG integration demonstrate the effectiveness and robustness of the proposed method.

distribution system state estimation↗

Assessing dynamic vegetation model parameter uncertainty across Alaskan arctic tundra plant communities

Abstract As the Arctic region moves into uncharted territory under a warming climate, it is important to refine the terrestrial biosphere models (TBMs) that help us understand and predict change. One fundamental uncertainty in TBMs relates to model parameters, configuration variables internal to the model whose value can be estimated from data. We incorporate a version of the Terrestrial Ecosystem Model (TEM) developed for arctic ecosystems into the Predictive Ecosystem Analyzer (PEcAn) framework. PEcAn treats model parameters as probability distributions, estimates parameters based on a synthesis of available field data, and then quantifies both model sensitivity and uncertainty to a given parameter or suite of parameters. We examined how variation in 21 parameters in the equation for gross primary production influenced model sensitivity and uncertainty in terms of two carbon fluxes (net primary productivity and heterotrophic respiration) and two carbon (C) pools (vegetation C and soil C). We set up different parameterizations of TEM across a range of tundra types (tussock tundra, heath tundra, wet sedge tundra, and shrub tundra) in northern Alaska, along a latitudinal transect extending from the coastal plain near Utqiaġvik to the southern foothills of the Brooks Range, to the Seward Peninsula. TEM was most sensitive to parameters related to the temperature regulation of photosynthesis. Model uncertainty was mostly due to parameters related to leaf area, temperature regulation of photosynthesis, and the stomatal responses to ambient light conditions. Our analysis also showed that sensitivity and uncertainty to a given parameter varied spatially. At some sites, model sensitivity and uncertainty tended to be connected to a wider range of parameters, underlining the importance of assessing tundra community processes across environmental gradients or geographic locations. Generally, across sites, the flux of net primary productivity (NPP) and pool of vegetation C had about equal uncertainty, while heterotrophic respiration had higher uncertainty than the pool of soil C. Our study illustrates the complexity inherent in evaluating parameter uncertainty across highly heterogeneous arctic tundra plant communities. It also provides a framework for iteratively testing how newly collected field data related to key parameters may result in more effective forecasting of Arctic change.

54 ENVIRONMENTAL SCIENCES↗

Enhancement of Distribution System State Estimation Using Pruned Physics-Aware Neural Networks

Realizing complete observability in the three-phase distribution system remains a challenge that hinders the implementation of classic state estimation algorithms. In this paper, a new method, called the pruned physics-aware neural network (P2N2), is developed to improve the voltage estimation accuracy in the distribution system. The method relies on the physical grid topology, which is used to design the connections between different hidden layers of a neural network model. To verify the proposed method, a numerical simulation based on one- year smart meter data of load consumptions for three-phase power flow is developed to generate the measurement and voltage state data. The IEEE 123-node system is selected as the test network to benchmark the proposed algorithm against the classic weighted least squares (WLS). Numerical results show that P2N2 outperforms WLS in terms of data redundancy and estimation accuracy.

distribution system state estimation↗

Enhancement of Distribution System State Estimation Using Pruned Physics-Aware Neural Networks: Preprint

Realizing complete observability in the three-phase distribution system remains a challenge that hinders the implementation of classical state estimation algorithms. In this paper, a new method so-called pruned physics-aware neural network (P2N2) is developed to improve the voltage estimation accuracy in the distribution system. The method relies on the physical grid topology, which is used to design the connections between different hidden layers of a neural network model. To verify the proposed method, a numerical simulation based on one-year smart meter data of load consumptions for threephase power flow is developed to generate the measurement and voltage state data. The IEEE 123 node system is selected as the test network to benchmark the proposed algorithm against the classical weighted least squares (WLS). Numerical results show that P2N2 outperforms WLS, in terms of data redundancy and estimation accuracy.

distribution systems state estimation↗

Distributed Generation Market Demand (dGen) model

The Distributed Generation Market Demand (dGen) model simulates customer adoption of distributed energy resources (DERs) for residential, commercial, and industrial entities in the United States or other countries through 2050. The dGen model can be used for identifying the sectors, locations, and customers for whom adopting DERs would have a high economic value, for generating forecasts as an input to estimate distribution hosting capacity analysis, integrated resource planning, and load forecasting, and for understanding the economic or policy conditions in which DER adoption becomes viable, and for illustrating sensitivity to market and policy changes such as retail electricity rate structures, net energy metering, and technology costs.

Array↗

Risk Considerations of Transitioning CO2-EOR Field to CO2 storage Field: Case Study

In the United States (U.S.), carbon dioxide (CO2) injection wells at EOR sites are currently regulated as Class II wells under the U.S. Environmental Protection Agency’s (EPA) Underground Injection Control (UIC) program while dedicated geological CO2 storage (GCS) wells are considered Class VI wells. A CO2-EOR facility considering a transitioning from tertiary oil recovery to injecting CO2 for the primary purpose of long-term storage is required to obtain a Class VI permit where this transition poses an increased risk to underground sources of drinking water. This study considers how transitioning operations from CO2-EOR to storage can impact reservoir plume and pressure transient in the storage envelope, and how these changes could impact area of review and potential unwanted fluid migration. We developed a case study to assess subsurface response and leakage risks associated with a representative, hypothetical operation in a carbonate reservoir. This reservoir transitions from tertiary hydrocarbon recovery to dedicated GCS. Reservoir simulations were run for a set of credible CO2-EOR scenarios to estimate distributions of fluids phases and pressures throughout the model domain after CO2 flooding as well as forecasting the behavior of the reservoir after the transition to a dedicated storage phase. The evolutions from all simulated scenarios were used as the basis for leakage risk quantification using the National Risk Assessment Partnership’s Open-Source Integrated Assessment Model (NRAP-Open-IAM) with a novel reduced-order model to estimate time-dependent leakage of CO2, brine, and hydrocarbon fluids through potentially leaky wells. Results include a description of reservoir response, an estimate of the areal extent that could potentially be impacted by leakage to underground sources of drinking water, and estimates of the magnitude of potential leakage. Considerations for dedicated storage injection well selection, injectivity, and injection scheme performance and potential leakage risk are presented, with implications for risk assessment of well transition discussed. This study presents a risk-based workflow for the Class II to Class VI well transition. Integrating credible numerical simulation of viable CO2-EOR to dedicated CO2 storage with quantitative risk assessment tools, such as the NRAP-Open-IAM, will provide a valuable means to devise operational scenarios and inform decision-making related to storage benefit, leakage risk, and liability. Presented at the SPE/AAPG/SEG Carbon Capture Utilization and Storage Conference in Houston, TX, March 11-13, 2024.

Liu, Guoxiang↗

Investigation of fuel film formation and soot emissions in a GDI engine during cold-start with Split-injection strategies

This study investigates the impact of fuel-film formation on engine-out soot emissions in a gasoline direct injection (GDI) engine under cold-start conditions. Split-injection strategies were applied by varying the number of injections, injection duration, and total fuel quantity to affect wall wetting and control the average in-cylinder equivalence ratio. A combined experimental and numerical approach was employed to analyze fuel-film deposition, combustion efficiency, and engine-out soot and unburnt hydrocarbons (UHC) emissions. In particular, fuel film distribution estimated by means of non-reacting, 3-D, computational fluid dynamics (CFD) simulations, together with experimentally measured soot data, were used to investigate fuel film formation and its role in soot generation. In the experiments, a skip-fired engine control strategy was applied to mimic the transient nature of engine cold-start operation. The results indicate that, under the same total number of injections, increasing the average in-cylinder equivalence ratio through longer injection durations improves combustion stability, as indicated by the decrease of the coefficient of variation of IMEP n (Net Indicated Mean Effective Pressure) from 6.0 % to 0.6 %. However, this strategy leads to higher soot emissions, which increased by nearly an order of magnitude, primarily due to enhanced wall-film formation. In contrast, increasing the number of injections while maintaining a constant equivalence ratio significantly impacts fuel-film deposition and, consequently, soot emissions, with a fivefold reduction of the measured engine-out soot, decreasing from 3.5 mg to 0.7 mg. A soot-film correlation was developed and achieved a high coefficient of determination (r 2 = 0.95) and was further extended to account for spark timing effects. These findings confirm the effectiveness of split-injection for avoiding wall film formation and soot emissions, and the critical role of fuel film in soot generation, supporting the hypothesis that pool fires play a crucial role in contributing to soot formation under these cold-start conditions. In conclusion, the study also indicates the value of a predictive soot-film correlation for developing cold-start emission control strategies.

Engine Cold-start↗

Comparing storm resolving models and climates via unsupervised machine learning

Global storm-resolving models (GSRMs) have gained widespread interest because of the unprecedented detail with which they resolve the global climate. However, it remains difficult to quantify objective differences in how GSRMs resolve complex atmospheric formations. This lack of comprehensive tools for comparing model similarities is a problem in many disparate fields that involve simulation tools for complex data. To address this challenge we develop methods to estimate distributional distances based on both nonlinear dimensionality reduction and vector quantization. Our approach automatically learns physically meaningful notions of similarity from low-dimensional latent data representations that the different models produce. This enables an intercomparison of nine GSRMs based on their high-dimensional simulation data (2D vertical velocity snapshots) and reveals that only six are similar in their representation of atmospheric dynamics. Furthermore, we uncover signatures of the convective response to global warming in a fully unsupervised way. Our study provides a path toward evaluating future high-resolution simulation data more objectively.

54 ENVIRONMENTAL SCIENCES↗

Joint modelling of astrophysical systematics for cosmology with LSST cosmic shear

ABSTRACT We present a novel framework for jointly modelling the weak lensing source galaxy redshift distribution and the intrinsic alignment (IA) of galaxies through a shared luminosity function (LF). In the context of a Rubin Observatory’s Legacy Survey of Space and Time (LSST) Year 1 and Year 10 cosmic shear analysis, we show that our novel approach produces cosmological parameter constraints which are comparable to standard methods, while offering more physical insight into IA and selection effects. We clarify the relationship between individual parameters of a Schechter LF and the redshift distribution of a magnitude-limited sample, showing the consequences of marginalizing over these parameters when modelling IAs in standard cosmic shear analyses. We explore the impact of the shape of the LF on the cosmic shear data vector, and we outline the potential of this method to naturally model selection functions in redshift distribution estimation. Although this work focuses on LSST cosmic shear, the proposed joint modelling framework is broadly applicable to weak lensing surveys.

Šarčević, Nikolina (ORCID:0000000173016415)↗

Arctic Shrub Expansion, Plant Functional Trait Variation, and Effects on Belowground Carbon Cycling (Final Technical Report)

Terrestrial ecosystems are undergoing dramatic changes in response to climate warming, and these changes are expected to feedback to the atmosphere, potentially altering the trajectory of future climate change. Feedbacks from Arctic ecosystems are a major concern because the Arctic is projected to warm significantly in the 21 st century and because >50% of global belowground organic carbon is stored in permafrost and overlying soils. Warming-driven release of this carbon could drastically increase atmospheric greenhouse gas concentrations and accelerate climate warming. Plant communities are also responding to warming, as evidenced by the widely documented increase in woody-shrub growth and “greening” across much of the Arctic tundra biome. This vegetation shift may offset or amplify warming by altering carbon cycling. The direction and magnitude of shrub effects remain highly uncertain, however, due to limited understanding of the consequences of shrub expansion for belowground carbon cycling and simplification of these relationships in models. The major shrubs expanding in the Arctic (Betula, Salix, and Alnus) vary widely with respect to aboveground and belowground traits (e.g., tissue production and chemistry, rooting depth, microbial symbionts), and may also exhibit substantial intraspecific variation in these traits in response to environmental conditions. Such variation is likely to have profound implications for soil carbon cycling. The overarching goal of this project was to improve process-based understanding of the influence of shrub expansion on carbon cycling to enable improved representation of carbon dynamics in ecosystem and Earth system models. We investigated how plant functional traits vary among shrub genera, respond to environmental conditions, and affect belowground carbon and nutrient cycling by quantifying relationships among functional traits and biogeochemical cycling along edaphic gradients nested within a climate gradient in the Alaskan tundra. We found consistent differences in leaf and root traits among shrub genera and between shrubs and a widespread sedge species, indicating diverse nutrient acquisition strategies and belowground impacts among different arctic shrubs. We also found striking differences in trait values among individuals within the same species or genera within sites. Soil parameters were more important than climate parameters for predicting size and leaf trait variation, and root trait responses were less dependent on climate overall. For all but one root trait, including parameters representing aboveground traits improved the predictive ability of models. These results demonstrate that tundra shrub traits vary considerably at local scales and soil factors drive this variation, especially belowground. Furthermore, leveraging information about aboveground traits and soil conditions can improve predictions of how belowground traits will respond to climate change. Despite these differences, soil carbon and nitrogen pools in the active layer did not vary among plots dominated by different shrub or sedge genera. Instead, pool sizes generally decreased from warmer to colder sites, consistent with a productivity gradient. Patterns of isotopic N composition indicate that shrubs tighten nitrogen cycling via nitrogen resorption or immobilization of shrub litter. Overall, these results suggest that further identifying the specific shrub genera in the tundra landscape will ultimately provide better predictions of belowground dynamics across the changing arctic. We also performed simulation experiments with the Terrestrial Ecosystem Model (TEM) incorporated in the Predictive Ecosystem Analyzer (PEcAn) framework, treats model parameters as probability distributions, estimates parameters based on a synthesis of available field data, and then quantifies both model sensitivity and uncertainty to a given parameter or suite of parameters. We performed simulations across different types of tundra, including shrub tundra. One key finding was that both model sensitivity and uncertainty to a given parameter could vary within the same type of tundra, but in a different geographical location, such as over the climate gradient of shrub tundra described above. We organized a special session at the annual meeting of the Ecological Society of America in August 2019 to disseminate our results, refine recommendations for model improvement, and initiate collaborations to implement these recommendations in existing models of tundra carbon dynamics at ecosystem to Earth system scales. Our results support DOE near-term priorities by providing mechanistic insights into the role of vegetation change in the terrestrial carbon cycle in a region that is inadequately represented in Earth system models. Current models reduce the complexity of Arctic vegetation to a small number of plant functional types (PFTs). This approach implicitly assumes that each PFT represents the average ecological function of its constituent species, thus ignoring the effects of trait variation on biogeochemical cycling and potentially leading to large uncertainty in the sign and magnitude of ecosystem feedbacks to climate. By quantifying variation of plant functional traits across broad gradients of climatic and edaphic conditions and elucidating the linkages of such variation with carbon and nutrient cycling, our results illustrate the need and create a foundation for further developing trait-based modeling approaches that allow the traits of PFTs to vary as a function of environmental conditions. These approaches should improve the capacity of simulation models to offer insights into ecosystem carbon dynamics associated with novel plant communities in a rapidly changing Arctic.

54 ENVIRONMENTAL SCIENCES↗

Efficient Network Partitioning: Application for Decentralized State Estimation in Power Distribution Grids: Preprint

Increase in the proliferation of DERs requires real-time situational awareness for efficient grid operations. State estimation plays an important role for real time control and management of the power grid. As the sensing infrastructure grows, aggregating and handling high volumes of data at a centralized location is extremely difficult. To address this challenge, this paper first proposes a novel and efficient hierarchical spectral clustering-based network partition algorithm followed by a decentralized compressive sensing (DCS) based state estimation. The applicability of the proposed network partitioning algorithm is tested on IEEE-123 bus, IEEE-8500 node, and a 6204-node distribution network. The results shows that the proposed approach efficiently divides the network into multiple sub-networks with the minimum edge connections among the neighbors. Then, we perform DCS-based state estimation on the 6204-node distribution network after dividing the network into 18 optimal partitions. Simulation results show that DCS-based state estimation recovers the system states with high accuracy and low complexity.

ADMM↗

Real-Time Distribution System State Estimation with Asynchronous Measurements

We report state estimation is a fundamental task in power systems. Although distribution systems are increasingly equipped with sensing devices and smart meters, measurements are typically reported at different rates and asynchronously; these aspects pose severe strains on workhorse state estimation algorithms, which are designed to process batches of data collected in a synchronous manner from all the measurement units. In this paper, we develop a novel state estimation algorithm to continuously update the estimate of the state based on measurements received in an asynchronous manner from measurement units. The synthesis of the algorithm hinges on a proximal-point type method, implemented in an online fashion, and capable of processing measurements received sequentially from sensors. A performance analysis is presented by providing bounds on the estimation error in terms of the mean and variance that hold at each iteration and asymptotically. The scheme is also compared with a more traditional Weighted Least Squares estimator that compensates for the lack of measurement data by using, as pseudo measurements, the measurement retrieved during a certain time window. Numerical simulations on the IEEE 37-bus feeder corroborate the analytical findings.

24 POWER TRANSMISSION AND DISTRIBUTION↗