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DOE OSTI2025

A Block-Structured Adaptive Mesh Framework to Solve Radiation Transfer Equation in Irregular Embedded Geometries

Radiation transport arises in various scientific, industrial, and medical fields, and understanding its effect in applications is needed to make accurate predictions, safety assessments and performance optimizations. Solving the Radiation Transport Equation (RTE) is challenging due to its integro-differential nature, which involves both differential and integral terms. The differential term describes the change in radiation intensity due to absorption and emission, while the integral term accounts for scattering. The accurate modeling of radiation is further complicated in many applications due to the complex, irregular geometries. Various methods exist for solving the RTE, including the zonal, Monte Carlo, spherical harmonics, discrete ordinates, and finite volume methods. Traditional mesh-based approaches, which rely on structured or unstructured meshes, struggle with irregular geometries due to: a) the difficulty of conforming structured grids to irregular domains, b) challenges in enforcing boundary conditions correctly, and c) the additional computational cost of unstructured mesh methods. This work presents a second-order accurate method for solving the RTE in irregular geometries. The radiation intensity is discretized using the finite-volume method in both spatial and angular directions on regular Cartesian grid blocks. Leveraging the block-structured adaptive mesh refinement (AMR) framework provided by AMReX, our method refines the grid locally to reduce spatial discretization error, ensuring a converged numerical solution while minimizing computational costs elsewhere. A two-stage deferred correction approach is employed: First, a first-order discretization on grid blocks is solved using an algebraic multigrid method in HYPRE. Second, a correction term is applied explicitly to achieve second-order accuracy. The correction term is calculated by approximating the radiation flux on cell faces using a Total Variation Diminishing (TVD) scheme. This approach ensures quick convergence of the multigrid method while preserving higher-order accuracy of the numerical solution. Irregular geometries are resolved as embedded boundaries (EB), resulting in both cut cells and regular cells. In cut cells, we modify the fluxes using face fractions and incorporate additional contributions from EB boundary conditions. To ensure higher-order convergence near the EB interface, the correction term is modified by interpolating the radiation intensity to fictitious ghost points. The implementation takes advantage of modern supercomputers by leveraging AMReX’sMPI/X parallelization strategy where X can be MPI or a GPU accelerator including CUDA, HIP and DPC++. We validate our solver using classical test cases, both with and without EB, demonstrating accuracy and efficiency. Additionally, we analyze the impact of adaptive mesh refinement on solution accuracy and computational cost, highlighting the advantages of our approach for high-resolution radiation transport simulations.

computational fluid dynamics (CFD)↗

DOE OSTI2021

A multiresolution adaptive wavelet method for nonlinear partial differential equations

We report the multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to solve partial differential equations (PDEs) with features evolving on a wide range of spatial and temporal scales. To meet these challenges, we present a multiresolution wavelet algorithm to solve PDEs with significant data compression and explicit error control. We discretize in space by projecting fields and spatial derivative operators onto wavelet basis functions. We provide error estimates for the wavelet representation of fields and their derivatives. Then, our estimates are used to construct a sparse multiresolution discretization which guarantees the prescribed accuracy. Additionally, we embed a predictor-corrector procedure within the temporal integration to dynamically adapt the computational grid and maintain the accuracy of the solution of the PDE as it evolves. We present examples to highlight the accuracy and adaptivity of our approach.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Pumped Storage Hydropower (PSH) FAST Commissioning Prize Technical Analysis

The US energy landscape has undergone major changes over the past 10 years and will continue to see significant changes in future decades as the power grid increases its reliance on variable renewable energy resources. Because of the inherent variability of these resources, renewable energy growth may require additional energy storage capacity to provide flexible load-following capabilities and other grid services that can quickly adjust to changes in energy demand and generation. Pumped storage hydropower (PSH)—one such energy storage technology—uses pumps to convey water from a lower reservoir to an upper reservoir for energy storage and releases water back to the lower reservoir via a powerhouse for hydropower generation. PSH facility pump and generation cycling often follows economic and energy demand conditions. Across the United States, 43 PSH facilities are in operation and 55 projects are in various permitting or licensing stages. Altogether, the 43 operational projects provide the wide majority (95%) of utility-scale electricity storage in the United States. These facilities also provide significant power and nonpower grid benefits, including large-scale electrical system reserve capacity, grid reliability support, and electricity supply-demand balancing through quick-response capabilities and operational flexibility. PSH systems can accomplish these at a scale (e.g., size) and cost that makes these systems highly attractive from a technical standpoint. Although these research concepts are still in their infancy, they demonstrate promising potential as future PSH energy storage technologies. Although PSH has many advantages, development in the United States has effectively stalled since the 1990s, partially because of the magnitude of project costs and financing interest during development and construction, the length of time from project investment until project revenue begins, permitting challenges, construction risks, competition from other storage technologies (e.g., batteries, hydrogen storage), and electricity market evolution and uncertainty. In short, the time, cost, and risk associated with modern PSH development have resulted in limited growth in the United States recently, despite the growing energy storage demand stemming from increased wind and solar power deployment. Technology innovation is needed to help reduce PSH commissioning time, cost, and risk, particularly during the post-licensing phase of project development. To address challenges facing the PSH industry and to improve PSH commissioning timelines, the US Department of Energy (DOE) Water Power Technologies Office (WPTO) initiated the PSH Furthering Advancements to Shorten Time to (FAST) Commissioning Prize project.

13 HYDRO ENERGY↗

DOE OSTI2022

Distributed Transient Safety Verification via Robust Control Invariant Sets: A Microgrid Application

Modern safety-critical energy infrastructures are increasingly operated in a hierarchical and modular control framework which allows for limited data exchange between the modules. In this context, it is important for each module to synthesize and communicate constraints on the values of exchanged information in order to assure system-wide safety. To ensure transient safety in inverter-based microgrids, we develop a set invariance-based distributed safety verification algorithm for each inverter module. Applying Nagumo's invariance condition, we construct a robust polynomial optimization problem to jointly search for safety-admissible set of control set-points and design parameters, under allowable disturbances from neighbors. We use sum-of-squares (SOS) programming to solve the verification problem and we perform numerical simulations using grid-forming inverters to illustrate the algorithm.

Bouvier, Jean-Baptiste H.↗

DOE OSTI2025

Rotated-Droop Control for Enhanced Stability and Power Decoupling in Microgrids With Complex Line Impedances

Classical droop control in microgrids predominantly assumes inductive line impedance, which simplifies implementation but causes power coupling and steady-state errors in systems with resistive and inductive lines. This paper proposes a rotated-droop control strategy for grid-forming inverters that reformulates the power equations by incorporating the magnitude and angle of the line impedance within a rotated reference frame. This method enhances the decoupling of active and reactive power dynamics without increasing complexity or requiring communication links. A small-signal state-space model was created to capture the dynamic behavior of the system under varying impedance parameters, preserving the original droop gains by rotating the power control structure. This allows eigenvalue-based stability analysis and enhances damping and transient performance. Simulation and experimental results validated the improved power-sharing performance, faster response, and robustness of the proposed method under different impedance conditions. This approach maintains the decentralized structure of the conventional droop control while enabling greater adaptability and scalability, making it suitable for modern inverter-based microgrids with dynamic topologies.

Campo-Ossa, Daniel Dario [Univ. of Puerto Rico, Ag↗

DOE OSTI2026

Study the Protection Improvements for a Weak Grid Area With High IBRs

NLR is collaborating with Florida Power & Light (FPL) and GE to investigate power system stability and protection reliability challenges in a weak-grid region with high penetration of inverter-based resources (IBRs). This presentation will primarily focus on the protection aspects of the study. We will share key insights from this real-world project, including best practices for developing high-fidelity fault study models, establishing a controller-hardware-in-the-loop (CHIL) platform for testing physical relays, identifying system-level protection challenges, and designing enhanced protection schemes to address those issues. Through this discussion, the audience will gain practical understanding of protection studies in IBR-dominated systems, the emerging challenges associated with reduced fault current and altered transient behavior, and effective mitigation strategies. In particular, we will highlight the critical importance of IBR compliance with IEEE 2800-2022 to ensure dependable and secure protection relay operation in modern transmission systems.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2023

Reinforcement Learning-based Output Structured Feedback for Distributed Multi-Area Power System Frequency Control

Load frequency control (LFC) is a key factor to maintain the stable frequency in multi-area power systems. As the modern power systems evolve from centralized to decentralized paradigm, LFC needs to consider the decentralized scheme that considers limited information from the information-exchange graph for the generator control of each interconnected area. This paper aims to solve a data-driven constrained LQR problem with mean-variance risk constraints and output structured feedback, and applies this framework to solve the LFC problem in multi-area power systems. By reformulating the constrained optimization problem into a minimax problem, the stochastic gradient descent max-oracle (SGDmax) algorithm with zero-order policy gradient (ZOPG) is adopted to find the optimal feedback gain from the learning, while guaranteeing the convergence. In addition, to improve the adaptation of the proposed learning method to new or varying models, we construct an emulator grid that approximates the dynamics of a physical grid and performs training based on this model. Once the feedback gain is obtained from the emulator grid, it is applied to the physical grid with a robustness test to check whether the controller from the approximated emulator applies to the actual system. Numerical tests show that the obtained feedback controller can successfully control the frequency of each area, while mitigating the uncertainty from the loads, with reliable robustness that ensures the adaptability of the obtained feedback gain to the actual physical grid.

Kwon, Kyung-bin↗

DOE OSTI2023

PNNL DSSE Docker Image

This is the docker image for Pacific Northwest National Laboratory's DSSE (distribution system state estimator) used for the demo of OEDI-SI platform. To support the operation of modern distribution systems, operators require real-time visibility into system states. Due to a lack of measurements and unbalanced operation, the state estimation in distribution systems is challenging. This submission is related to an OEDI-SI use case which demonstrates the application of an extended Kalman filter-based state estimator on an IEEE-123 bus system. The state estimator uses the measurements to generate voltage estimates for the system.

Array↗

NASA NTRS2006

Computational Analysis of a Chevron Nozzle Uniquely Tailored for Propulsion Airframe Aeroacoustics

A computational flow field and predicted jet noise source analysis is presented for asymmetrical fan chevrons on a modern separate flow nozzle at take off conditions. The propulsion airframe aeroacoustic asymmetric fan nozzle is designed with an azimuthally varying chevron pattern with longer chevrons close to the pylon. A baseline round nozzle without chevrons and a reference nozzle with azimuthally uniform chevrons are also studied. The intent of the asymmetric fan chevron nozzle was to improve the noise reduction potential by creating a favorable propulsion airframe aeroacoustic interaction effect between the pylon and chevron nozzle. This favorable interaction and improved noise reduction was observed in model scale tests and flight test data and has been reported in other studies. The goal of this study was to identify the fundamental flow and noise source mechanisms. The flow simulation uses the asymptotically steady, compressible Reynolds averaged Navier-Stokes equations on a structured grid. Flow computations are performed using the parallel, multi-block, structured grid code PAB3D. Local noise sources were mapped and integrated computationally using the Jet3D code based upon the Lighthill Acoustic Analogy with anisotropic Reynolds stress modeling. In this study, trends of noise reduction were correctly predicted. Jet3D was also utilized to produce noise source maps that were then correlated to local flow features. The flow studies show that asymmetry of the longer fan chevrons near the pylon work to reduce the strength of the secondary flow induced by the pylon itself, such that the asymmetric merging of the fan and core shear layers is significantly delayed. The effect is to reduce the peak turbulence kinetic energy and shift it downstream, reducing overall noise production. This combined flow and noise prediction approach has yielded considerable understanding of the physics of a fan chevron nozzle designed to include propulsion airframe aeroacoustic interaction effects.

Massey, Steven J.↗

DOE OSTI2022

A low-cost centralized HVAC control system solution for energy savings, load shedding, and improved maintenance

University campuses rely on centralized controls for managing and optimizing complex HVAC systems in larger buildings. However, most campuses also have many smaller buildings with packaged HVAC systems controlled by a stand-alone thermostat. Even when these distributed and often overlooked systems have modern programmable thermostats, they cannot be centrally monitored or controlled, and they are typically not programmed adequately. This paper describes the implementation of a low-cost centralized control solution for these systems serving smaller campus buildings, mostly under 5,000 sf and representative of light commercial spaces. Thanks to advances in technology spurred by residential and commercial IoT developments, simple networked thermostat solutions exist that can easily replace original thermostats, and, connect these systems to a web-based portal for monitoring and control. We show that, with small customizations, these platforms can be integrated into facility management workflows. Beyond the energy savings potential from improved scheduling and closer management of these systems, there are significant advantages for maintenance crews since these systems can now be monitored on smart phones or tablets. A grid-responsive load-shedding program has also been implemented for additional cost savings. The networked thermostats can also be connected to additional systems such as economizer controls for improved ventilation management and energy savings. With data from these systems integrated centrally, it can also be used for improved analytics and fault detection. A toolkit has been developed to share the program with other campuses, whether for energy savings, improved management of ventilation, or a more proactive maintenance approach.

Fauchier-Magnan, Nicolas↗

NASA NTRS2008

NASA's Modern Era Retrospective-Analysis for Research and Applications (MERRA): Early Results and Future Directions

This talk will review the status and progress of the NASA/Global Modeling and Assimilation Office (GMAO) atmospheric global reanalysis project called the Modern Era Retrospective-Analysis for Research and Applications (MERRA). An overview of NASA's emerging capabilities for assimilating a variety of other Earth Science observations of the land, ocean, and atmospheric constituents will also be presented. MERRA supports NASA Earth science by synthesizing the current suite of research satellite observations in a climate data context (covering the period 1979-present), and by providing the science and applications communities with of a broad range of weather and climate data with an emphasis on improved estimates of the hydrological cycle. MERRA is based on a major new version of the Goddard Earth Observing System Data Assimilation System (GEOS-5), that includes the Earth System Modeling Framework (ESMF)-based GEOS-5 atmospheric general circulation model and the new NOAA National Centers for Environmental Prediction (NCEP) unified grid-point statistical interpolation (GST) analysis scheme developed as a collaborative effort between NCEP and the GMAO. In addition to MERRA, the GMAO is developing new capabilities in aerosol and constituent assimilation, ocean, ocean biology, and land surface assimilation. This includes the development of an assimilation capability for tropospheric air quality monitoring and prediction, the development of a carbon-cycle modeling and assimilation system, and an ocean data assimilation system for use in coupled short-term climate forecasting.

Schubert, Siegfried↗

NASA NTRS1990

Unsteady transonic Navier-Stokes computations for an oscillating wing using single and multiple zones

Modern jet transports and maneuvering tactical fighters operating in the transonic regime often give rise to time-dependent fluid physics that interact with flexible structural components, e.g., vortical flow, shocks, and separation. Efficient computational fluid dynamic (CFD) methods are required to study such computationally intensive problems. In this work a numerical method is presented to address this problem. Time-dependent, compressible, Navier-Stokes equations are used to simulate unsteady transonic flow about a three-dimensional rigid wing undergoing a forced periodic motion in angle of attack. An efficient, implicit, diagonal algorithm is utilized because of its low operation count per time step compared to other methods that solve systems of block matrix equations. The formal time accuracy is addressed theoretically and demonstrated numerically by comparison of computational results with experimental data. A zonal grid approach, capable of treating complex geometries, is presented and its time accuracy is demonstrated by comparing a zonal computation with a single grid computation and experimental data.

Chaderjian, Neal M.↗

NASA NTRS2019

Assimilation of Satellite-Based Snow Cover and Freeze/Thaw Observations Over the High Mountain Asia

Toward qualifying hydrologic changes in the High Mountain Asia (HMA) region, this study explores the use of a hyper-resolution (1 km) land data assimilation (DA)framework developed within the NASA Land Information System using the Noah Multi-parameterization Land Surface Model (Noah-MP) forced by the meteorological boundary conditions from Modern-Era Retrospective analysis for Research and Applications, Version 2 data. Two different sets of DA experiments are conducted:(1) the assimilation of a satellite-derived snow cover map (MOD10A1) and (2) the assimilation of the NASA MEaSUREs landscape freeze/thaw product from 2007 to 2008. The performance of the snow cover assimilation is evaluated via comparisons with available remote sensing-based snow water equivalent product and ground-based snow depth measurements. For example, in the comparison against ground-based snow depth measurements, the majority of the stations (13 of 14) show slightly improved goodness-of-fit statistics as a result of the snow DA, but only four are statistically significant. In addition, comparisons to the satellite-based land surface temperature products (MOD11A1 and MYD11A1) show that freeze/thaw DA yields improvements (at certain grid cells) of up to 0.58 K in the root-mean-square error (RMSE) and 0.77K in the absolute bias (relative to model-only simulations). In the comparison against three ground-based soil temperature measurements along the Himalayas, the bias and the RMSE in the 0-10 cm soil temperature are reduced (on average) by 10 and 7%,respectively. The improvements in the top layer of soil estimates also propagate through the deeper soil layers, where the bias and the RMSE in the 10-40 cm soil temperature are reduced (on average) by 9 and 6%, respectively. However, no statistically significant skill differences are observed for the freeze/thaw DA system in the comparisons against ground-based surface temperature measurements at mid-to-low altitude. Therefore, the two proposed DA schemes show the potential of improving the predictability of snow mass, surface temperature, and soil temperature states across HMA, but more ground-based measurements are still required, especially at high-altitudes, in order to document a more statistically significant improvement as a result of the two DA schemes.

High Mountain Asia↗

DOE OSTI2026

The 2025 Evaluation of Experimental Thermonuclear Reaction Rates (ETR25)

This work describes the formalism for estimating thermonuclear reaction rates for astrophysical applications, emphasizing modern statistical approaches such as Monte Carlo sampling and Bayesian models. We discuss related topics including the calculation of resonance energies from nuclear Q values, indirect estimates of particle partial widths, and matching of reaction rates at elevated temperatures to statistical model results. We have evaluated available experimental data on cross sections, resonance energies and strengths, partial widths, lifetimes, spin-parities, and spectroscopic factors. Based on these results, we have estimated numerical values of 78 experimental charged-particle thermonuclear reaction rates for target nuclei in the A = 2–40 mass region, for temperatures ranging from 1 MK to 10 GK. For each reaction, three rate values are provided: low, median, and high, corresponding to the 16th, 50th, and 84th percentiles, respectively, of the cumulative reaction rate probability density distribution. Additionally, we present the factor uncertainty of each rate at each temperature grid point. These results enable users to sample the reaction rate probability density in nucleosynthesis calculations, facilitating uncertainty estimates of nuclidic abundances. The rates presented here refer to their laboratory values. For use in stellar model simulations, these values need to be corrected for the effects of thermal excitations of the interacting nuclei. For each reaction, we include graphs that illustrate the fractional contributions to the overall reaction rate along with the associated uncertainty. These visuals are designed to assist both stellar modelers and nuclear experimentalists by identifying the primary sources of rate uncertain=^texttx);ty at specific stellar temperatures. A graphical comparison with earlier Monte Carlo rates is also provided.

Nuclear astrophysics↗

NASA NTRS2014

A Modular Approach to Model Oscillating Control Surfaces Using Navier Stokes Equations

The use of active controls for rotorcraft is becoming more important for modern aerospace configurations. Efforts to reduce the vibrations of helicopter blades with use of active-controls are in progress. Modeling oscillating control surfaces using the linear aerodynamics theory is well established. However, higher-fidelity methods are needed to account for nonlinear effects, such as those that occur in transonic flow. The aeroelastic responses of a wing with an oscillating control surface, computed using the transonic small perturbation (TSP) theory, have been shown to cause important transonic flow effects such as a reversal of control surface effectiveness that occurs as the shock wave crosses the hinge line. In order to account for flow complexities such as blade-vortex interactions of rotor blades higher-fidelity methods based on the Navier-Stokes equations are used. Reference 6 presents a procedure that uses the Navier-Stokes equations with moving-sheared grids and demonstrates up to 8 degrees of control-surface amplitude, using a single grid. Later, this procedure was extended to accommodate larger amplitudes, based on sliding grid zones. The sheared grid method implemented in EulerlNavier-Stokes-based aeroelastic code ENS AERO was successfully applied to active control design by industry. Recently there are several papers that present results for oscillating control surface using Reynolds Averaged Navier-Stokes (RANS) equations. References 9 and 10 report 2-D cases by filling gaps with overset grids. Reference 9 compares integrated forces with the experiment at low oscillating frequencies whereas Ref. 10 reports parametric studies but with no validation. Reference II reports results for a 3D case by modeling the gap region with a deformed grid and compares force results with the experiment only at the mid-span of flap. In Ref. II grid is deformed to match the control surface deflections at the section where the measurements are made. However, there is no indication in Ref. II that the gaps are explicitly modeled as in Ref. 6. Computations using overset grids are reported in Ref. 12 for a case by adding moving control surface to an existing blade but with no validation either with an experiment or another computation.

Navier↗

DOE OSTI2025

Hawaii Threat Brief

The Digital Energy Transformation is redefining how power systems operate, integrating physical infrastructure with digital technologies to enhance efficiency, visibility, and resilience. For islanded and digitally modernizing systems like Hawai‘i’s, this shift presents both new opportunities and evolving challenges in cybersecurity, supply chain assurance, and environmental resilience. This brief provides an overview of critical infrastructure dependencies and systemic risks associated with increasing digital integration. It summarizes recent energy-sector threat activity and case studies that illustrate adversary tactics and supply chain vulnerabilities, and identifies pathways to strengthen resilience.

25 - ENERGY STORAGE↗

NASA NTRS2016

Extremely Severe Space Weather and Geomagnetically Induced Currents in Regions with Locally Heterogeneous Ground Resistivity

Large geomagnetically induced currents (GICs) triggered by extreme space weather events are now regarded as one of the serious natural threats to the modern electrified society. The risk is described in detail in High-Impact, Low-Frequency Event Risk, A Jointly-Commissioned Summary Report of the North American Electric Reliability Corporation and the US Department of Energy's November 2009 Workshop, June 2010. For example, the March 13-14,1989 storm caused a large-scale blackout affecting about 6 million people in Quebec, Canada, and resulting in substantial economic losses in Canada and the USA (Bolduc 2002). Therefore, European and North American nations have invested in GIC research such as the Solar Shield project in the USA (Pulkkinen et al. 2009, 2015a). In 2015, the Japanese government (Ministry of Economy, Trade and Industry, METI) acknowledged the importance of GIC research in Japan. After reviewing the serious damages caused by the 2011 Tohoku-Oki earthquake, METI recognized the potential risk to the electric power grid posed by extreme space weather. During extreme events, GICs can be concerning even in mid- and low-latitude countries and have become a global issue.

Fujita, Shigeru↗

DOE OSTI2025

Spacetime pq theory for AC and DC electric power systems

The 50/60 Hz alternating current (AC) electric power has been the standard and most flexible energy source powering our modern societies for one and a half centuries since the war of the currents: AC versus direct current (DC). A reactive power concept that was introduced at the beginning of the AC power was very useful for circuit/system analysis, design, control, optimization, and ultimately for more efficient and stable generation, transmission, distribution, and consumption. The initial reactive power theory was based on single-phase sinusoidal AC power to capture inductive and capacitive power that yields to net-zero average power over one fundamental cycle. Soon it was expanded to non-sinusoidal AC power and finally to instantaneous three-phase AC power. However, these reactive power theories remain separate and limited to special cases and have never been consolidated and made valid to all cases. Today, more widespread adoption of power electronics and renewable energy is bringing back DC power into the electric grids. The reactive power concept has never been applied to DC power systems. There is no reactive power in DC power systems according to the existing reactive power theories. Do DC power systems really have no reactive power? Capacitors and inductors are widely used in DC just like in AC power systems. Are they not reactive power components? Why are they different from their AC counterparts? Furthermore, are batteries active or reactive power components? What about active devices like power converters (or inverters) with AC (or DC) on one side and DC (or AC) on the other? Do they generate or consume reactive power? Finally, what about AC and DC hybrid power systems? How to define reactive power in such a complex power system that has a multitude of loads, buses, and sources? Is there reactive power between any two loads, any two buses, or any two sources in a power system and what is the total reactive power in such a complex power system as a whole? As the motivation and goal of this paper to answer the above basic questions, to unify the existing AC reactive power theories and to ultimately provide theoretical and insightful guidance for system analysis, design, control, efficiency, optimization, and operation of complex power systems, a concept of spacetime (both spatial and temporal) active and reactive power (pq) theory—the spatiotemporal aspect of active and reactive power—is developed for both AC and DC power systems. The theoretical definitions and physical meanings of the spacetime reactive power will be developed, and real applications and thought experiments/cases/exercises will be explored and discussed. The developed mathematics to define the active (or real) and reactive (or imaginary) power— p and q respectively by dot (scalar) and cross (vector) products of multi-dimension spacetime vectors and time-space mapping principle/law can have some fundamental implications as well.

24 POWER TRANSMISSION AND DISTRIBUTION↗