Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “electric load”

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 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Low-temperature solar thermal-power systems for residential electricity supply under various seasonal and climate conditions

In this work, the performance of low-temperature (<100 degrees C) solar thermal-power systems to satisfy residential electric loads was analyzed. The solar-driven system was designed to provide a fraction of the total electricity demand in a complementary operation with the electric grid. The analysis was conducted for an coperation during seven days each season, considering real solar and climate variables and residential loads at different climate zones in the United States. The efficiency of the system strongly depends on the solar radiation profile and the ambient temperature. Maximum efficiencies of around 9.5% were obtained in the cold and marine climate zones due to the high solar energy input and low heat dissipation temperatures. In these two zones, the system could supply more than 98% of the electricity demand all seasons. At mixed-humid and hot-humid regions, the system supplied around 50% of the electric load in three of the four seasons, but it only supplies about 27% of the electricity needs in the mixed-humid zone during summer and hot-humid zone during winter. The effect of the solar collector field area and the tank volume was also analyzed. In general, larger solar fields positively impact the efficiency. However, the impact of the tank volume varies depending on the solar radiation profile and the load requirements. Average efficiencies for the seven-day operation can be larger than 6% with a proper selection of the solar collector area and tank volume for an Organic Rankine Cycle with a capacity of about 2.6 kW. Finally, an economic analysis of the system was conducted, and the results were compared with a solar PV + battery system of similar capacity. It is expected that the cost for the analyzed solar-thermal system decreases in the coming years with the increased interest on low temperature applications.

14 SOLAR ENERGY↗
DOE OSTI2024

Deep-Learning-Based Multi-Timescale Load Forecasting in Buildings: Opportunities and Challenges from Research to Deployment

Electricity load forecasting for buildings and campuses is becoming increasingly important as the penetration of distributed energy resources (DERs) grows. Efficient operation and dispatch of DERs require reasonably accurate predictions of future energy consumption in order to conduct near-real-time optimized dispatch of on-site generation and storage assets. Electric utilities have traditionally performed load forecasting for load pockets spanning large geographic areas, and therefore, forecasting has not been a common practice by buildings and campus operators. Given the growing trends of research and prototyping in the grid-interactive efficient buildings domain, characteristics beyond simple algorithm forecast accuracy are important in determining the algorithm's true utility for smart buildings. Other characteristics include the overall design of the deployed architecture and the operational efficiency of the forecasting system. In this work, we present a deep-learning-based load forecasting system that predicts the building load at 1-hour intervals for 18 hours in the future. We also discuss challenges associated with the real-time deployment of such systems as well as the research opportunities presented by a fully functional forecasting system that has been developed within the National Renewable Energy Laboratory's Intelligent Campus program.

building load forecasting↗
DOE OSTI2025

Packages of Distributed Energy Technologies Demonstrating Demand Flexibility at Community Scale

The combination of increased electric load growth across all sectors, deferred electrical infrastructure investment, and other factors resulting in variable electric power supply, has created technical challenges to maintaining a resilient and reliable grid. Many federal, regional, and local efforts are in play to modernize the electric grid, including advancing building technologies and distributed energy resources (DERs) that are utilizing smarter controls to become responsive to both occupant and grid needs. This report reviews ten pilot projects demonstrating how groups of buildings combined with behind-the-meter (BTM) DERs such as electric vehicle (EV) charging, battery storage, flexible HVAC and domestic hot water systems, and photovoltaic systems can reliably and cost effectively provide grid services. Each of the ten pilot projects aim to deliver both energy efficiency and demand flexibility (DF) while supporting load growth. The ten demonstration teams are piloting flexible DER packages across diverse communities of residential and commercial buildings to address a variety of regional grid needs. The outcomes of these pilot projects will be used to inform future scaling through utility program development. This paper characterizes the ten teams, showcasing the decision-making process used by each group to develop their packages (Section 2), the grid services they plan to deliver (Section 3), the types of DER packages selected for deployment within building sectors (Section 4) and trends between building sector, DER types, and grid services In order to achieve community scale benefits, the pilot projects must utilize aggregated control mechanisms for coordinating buildings and DERs together. Several types of coordinated control architectures have evolved amongst the teams, influenced by use type, existing market conditions, and integration type. Three coordinated controls architectures have been characterized, highlighting their use cases, benefits, challenges, and tradeoffs in their design. These insights can aid utilities, control vendors, and developers in scaling community-level energy systems (Paul, 2024). Ultimately, the technology packages selected by the ten teams will be coordinated to provide power system services, also known as grid services. Insights from these demonstrations will be useful for grid operators, regulators, aggregators and other stakeholders as they look to deploy demand flexible resources as grid services in the future. The grid services that each team is targeting for demonstration are described in Section 3 and Section 4. Methods for evaluating the grid services have been described in the paper Metrics for Evaluating Grid Service Provision from Communities of Grid-interactive and Efficient Buildings and other DER (MacDonald, 2023). To identify technology packages for demonstration, Section 2 shows that project teams used a range of analysis approaches, including building energy modeling, AMI data analysis, cost-benefit frameworks, and utility pilot data. Some teams emphasized technical modeling to quantify grid impacts and demand reduction potential, while others prioritized economic evaluations, stakeholder input, or exploratory pilots to inform deployment decisions. This diversity reflects the need to tailor selection methods to project goals, available data, and organizational context. Section 5 discusses trends between the DER technologies deployed and the grid service provisions from each team. Residential buildings (multifamily and single family) lean towards technologies that enhance energy efficiency (e.g. weatherization upgrades, smart thermostats) and onsite power generation integration (e.g. solar PV). Commercial building demonstrations prioritize technologies that ensure operational reliability (e.g. battery storage) and centralized energy management systems and optimization solutions. Teams that are deploying controllable storage-based technologies are more likely to provide grid services that require a near real-time response. Teams incorporating load shifting technologies like smart thermostats with HEMs are likely to include energy markets participation and customer bill management offerings. Campus demonstrations are adopting diverse sets of DERs to emphasize renewable generation, paired with centralized control. This section also describes technologies that were considered during project planning but ultimately excluded from final deployment. These demonstrations reveal that effective DER package design should be tailored to building type, customer segment, and construction vintage. Multifamily buildings benefit from centralized HVAC upgrades and supervisory controls, while single-family homes are well-suited for individualized technologies like solar, storage, and smart home energy monitors. Commercial and campus settings prioritize EMIS integration and load optimization. New construction enables cost-effective integration of DER-ready infrastructure, whereas retrofits require deployments aligned with owner and tenant value streams. For utility program planners, early coordination with developers and building owners, paired with segmented and modular program offerings, can improve adoption, scalability, and grid impact.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

MPC solution for optimal load shifting for buildings with ON/OFF staged packaged units: Experimental demonstration, and lessons learned

Small and medium-sized commercial buildings (SMCB) are significant demand response resources, and it is important to develop grid-responsive control algorithms that exploit those resources and create financial benefits for building owners and HVAC service providers. Furthermore, unlike large-sized commercial buildings, there is an opportunity to have universally applicable control solutions for many SMCBs since those buildings have a consistent HVAC system configuration: SMCBs are commonly served by multiple-staged air conditioning units controlled by their own thermostats. Despite the demand response potential and scalability, however, very few control solutions are available for SMCBs. Typical model predictive control (MPC) and heuristic control approaches for cooling load shifting that lower thermostat setpoints before an electric price jump are suitable mainly for large-sized commercial buildings where a continuous capacity modulation is possible, e.g., via dampers in variable air volume terminal units. However, those approaches can cause undesired, high peaks for SMCBs due to the nature of ON/OFF unit staging and narrow thermostat deadbands. This could discourage the use of advanced grid-responsive controls for SMCBs due to the concern of high demand charges, and has to be resolved. This paper presents a MPC solution that overcomes this challenge. It has a hierarchical MPC structure where an upper level MPC is responsible for electrical load shifting in response to an electric price signal while a lower level MPC is responsible for coordinating compressor stages to eliminate unnecessary peaks and follows the setpoints determined by the upper level MPC. In this work, two one-month, comprehensive laboratory tests have been carried out to demonstrate load shifting and cost savings for the algorithm. Interesting trade-offs between energy efficiency and load flexibility were observed and are discussed, and lessons learned for applying MPCs for SMCBs are also presented.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2023

A Siamese CNN + KNN-Based Classification Framework for Non-intrusive Load Monitoring

Through the development of smart grids, programs such as demand side response, have been presented as auxiliary services to the real-time operation of distributed networks. In order to provide consumers information on their energy consumption, so that a modulation in consumption is possible, non-intrusive load monitoring has been introduced as an solution to this pattern recognition problem. Non-intrusive load monitoring enables the modeling of electrical loads connected to the low-voltage system, considering only a single measurement point. Presented state-of-the-art solutions though, consider availability of data as well as representation of all possible classes of the environment. This is of course a most conservative hypothesis, since in real-life applications availability of such data is much difficult, as well as the dynamic behavior of models is implicitly evolving in time. Here, a framework that uses neural Siamese networks with k-nearest neighbor clustering is presented toward non-intrusive load monitoring. Online learning feature is implemented, which relaxes the hypothesis of data requirements as well addresses the evolving nature of load profile. k-nearest clustering allows nonlinear characteristic space modelling. Test results using synthetics and real-life data show that the solution, besides obtaining a good generalizability in the classification, also obtained results with an accuracy of 95.77%.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2022

Innovative Technologies for a Low Carbon Electricity System

The electricity sector represents the centerpiece of decarbonization pathways for the state. Decrease in the cost of renewable electricity generation, combined with ample solar energy, wind and other renewable resources, presents a realistic way to achieve electricity generation that is nearly free of CO 2 emissions by mid-century. Expansion of renewable electricity supply could allow replacement of many CO 2 -emitting technologies with ones that use electricity—in transportation, buildings, and possibly industry. Key elements of the path for California’s electricity sector are: restrain electricity demand through higher efficiency, rapidly expand renewable electricity generation, develop electricity storage to complement renewable electricity, manage flexible electricity loads for a low-carbon electricity system, electrify where appropriate to reduce CO 2 emissions, and maintain reliable and resilient electricity supply. This report provides an overview of a multitude of innovative technologies in each of the above areas that have the potential to help the state meet its decarbonization goals, while lowering costs and promoting greater reliability. The information presented provides a portrait of the landscape of technology innovation that can help policymakers, state agencies, and interested parties develop strategies to meet the state’s goals and to target efforts to support and nurture technology innovation.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2021

Energy use of residential safety, security, and health devices

Miscellaneous electrical loads (MELs) are responsible for a significant fraction of energy consumption in buildings. This paper addresses one category of MELs: Safety, Security, and Health Devices (SSHDs). Common SSHDs include: electrical life safety equipment, smoke alarms, radon mitigation fans, and home oxygen concentrators. The installation or use of these devices is dictated by building codes, health providers, insurance companies, and other entities—none of which would ordinarily consider energy efficiency a priority. For this project, the most important residential devices were explored in terms of their governing regulations, functions, technologies, and energy use. The power consumption of 41 life safety SSHDs were measured. Individual power consumption of life safety devices is very small but many are typically required in every home. Devices performing the same functions have wide ranges in power use. Opportunities for reducing SSHD energy use were investigated, the results are summarized, and recommendations are made for further actions that could be taken to reduce SSHD energy use. Finally, SSHDs identified in this paper are responsible for at least one percent of current U.S. residential electricity consumption, but that number will climb steadily as existing buildings are upgraded and new types of SSHDs appear.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2020

Development of Real-Time System Identification to Detect Abnormal Operations in a Gas Turbine Cycle

Here, we present a novel online system identification methodology for monitoring the performance of power systems. This methodology was demonstrated in a gas turbine recuperated power plant designed for a hybrid configuration. A 120-kW Garrett microturbine modified to test dynamic control strategies for hybrid power systems designed at the National Energy Technology Laboratory (NETL) was used to implement and validate this online system identification methodology. The main component of this methodology consists of an empirical transfer function model implemented in parallel to the turbine speed operation and the fuel control valve, which can monitor the process response of the gas turbine system while it is operating. During fully closed-loop operations or automated control, the output of the controller, fuel valve position, and the turbine speed measurements were fed for a given period of time to a recursive algorithm that determined the transfer function parameters during the nominal condition. After the new parameters were calculated, they were fed into the transfer function model for online prediction. The turbine speed measurement was compared against the transfer function prediction, and a control logic was implemented to capture when the system operated at nominal or abnormal conditions. To validate the ability to detect abnormal conditions during dynamic operations, drifting in the performance of the gas turbine system was evaluated. A leak in the turbomachinery working fluid was emulated by bleeding 10% of the airflow from the compressor discharge to the atmosphere, and electrical load steps were performed before and after the leak. This tool could detect the leak 7 s after it had occurred, which accounted for a fuel flow increase of approximately 15.8% to maintain the same load and constant turbine speed operations.

algorithms↗
DOE OSTI2020

AlphaBuilding - Synthetic Buildings Operation Dataset

This is a synthetic building operation dataset which includes HVAC, lighting, miscellaneous electric loads (MELs) system operating conditions, occupant counts, environmental parameters, end-use and whole-building energy consumptions at 10-minute intervals. The data is created with 1395 annual simulations using the U.S. DOE detailed medium-sized reference office building, and 30 years' historical weather data in three typical climates including Miami, San Francisco, and Chicago. Three energy efficiency levels of the building and systems are considered. Assumptions regarding occupant movements, occupants' diverse temperature preferences, lighting, and MELs are adopted to reflect realistic building operations. A semantic building metadata schema - BRICK, is used to store the building metadata. The dataset is saved in a 1.2 TB of compressed HDF5 file. This dataset can be used in various applications, including building energy and load shape benchmarking, energy model calibration, evaluation of occupant and weather variability and their influences on building performance, algorithm development and testing for thermal and energy load prediction, model predictive control, policy development for reinforcement learning based building controls.

AlphaBuilding↗
DOE OSTI2021

Endpoint Use Efficiency Comparison for AC and DC Power Distribution in Commercial Buildings

Advances in power electronics and their use in Miscellaneous Electric Loads (MELs) in buildings have resulted in increased interest in using low-voltage direct current (DC) power distribution as a replacement for the standard alternating current (AC) power distribution in buildings. Both systems require an endpoint converter to convert the distribution system voltage to the MELs voltage requirements. This study focused on the efficiency of these endpoint converters by testing pairs of AC/DC and DC/DC power converters powering the same load profile. In contrast to prior studies, which estimated losses based on data sheet efficiency and rated loads, in this study, we used part load data derived from real-world time-series load measurements of MELs and experimentally characterized efficiency curves for all converters. The measurements performed for this study showed no systematic efficiency advantage for commercially available DC/DC endpoint converters relative to comparable, commercially available AC/DC endpoint converters. For the eight appliances analyzed with the pair of converters tested, in 50%, the weighted energy efficiency of the DC/DC converter was higher, while, for the other 50%, the AC/DC converter was. Additionally, the measurements indicated that the common assumption of using either data sheet efficiency values or efficiency at full load may result in substantial mis-estimates of the system efficiency.

AC/DC converters↗
DOE OSTI2020

Simulating dispatchable grid services provided by flexible building loads: State of the art and needed building energy modeling improvements

End-use electrical loads in residential and commercial buildings are evolving into flexible and cost-effective resources to improve electric grid reliability, reduce costs, and support increased hosting of distributed renewable generation. This article reviews the simulation of utility services delivered by buildings for the purpose of electric grid operational modeling. We consider services delivered to (1) the high-voltage bulk power system through the coordinated action of many, distributed building loads working together, and (2) targeted support provided to the operation of low-voltage electric distribution grids. Although an exhaustive exploration is not possible, we emphasize the ancillary services and voltage management buildings can provide and summarize the gaps in our ability to simulate them with traditional building energy modeling (BEM) tools, suggesting pathways for future research and development.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2021

Improving the Accuracy of Clustering Electric Utility Net Load Data using Dynamic Time Warping

Identifying patterns in electric utility net load data in a time-series format is very useful in preparing the operation for next day. Machine learning algorithms have been used in other domains and those concepts are applied in this paper on real-world net load measurement data. Clustering is the practice of grouping data with similar characteristics as determined by the distance measure. The K-means clustering algorithm is utilized here with actual electric utility data. The paper uses the standard distance measure, Euclidean distance (ED), and compares its performance against the dynamic time warping (DTW) measure. An actual case study with real data is presented, and DTW distance measure-based method observed to result better accuracy compared to the ED based method for substation net load measurements predominantly with residential customers.

clustering↗
DOE OSTI2022

A multi-level load shape clustering and disaggregation approach to characterize patterns of energy consumption behavior

This study presents representative electrical load shapes, disaggregated to the end-use level, for over 5000 customer clusters across California’s residential, commercial, industrial and agricultural sectors. We developed a novel, multi-level load shape clustering approach for residential and commercial sectors leveraging interval meter data for over 350,000 California utility customers collected as a part of the Phase 4 California Demand Response (DR) Potential Study. The clustering approach allowed us to identify typical consumption patterns and categorize customers based on their daily load shape displayed throughout the year. For example, we were able to identify customers with particular energy technologies such as electric vehicles and rooftop solar, as well as building occupancy types such as restaurants, grocery stores and even unoccupied buildings, based solely on whole-building interval data. We then combined the load shape-based clusters with other customer information including building type, climate, geographical area, total consumption and low-income status, to create a set of customer clusters based on both demographics and usage patterns. Total cluster electricity demand was then disaggregated into a wide variety of end-uses using weather normalization and other publicly available end-use load shape datasets. The resulting disaggregated cluster load shapes will be released in anonymized form as part of the Phase 4 DR Potential Study. They will have wide-ranging applications in energy research and policy analysis, including estimation of energy efficiency (EE) and DR potential on the end-use level, time-dependent valuation of EE savings, building stock modeling, and developing customer targeting strategies for EE and DR programs.

Murthy, Samanvitha↗
DOE OSTI2020

Synthetic residential load models for smart city energy management simulations

The ability to control tens of thousands of residential electricity customers in a coordinated manner has the potential to enact system-wide electric load changes, such as reduce congestion and peak demand, among other benefits. To quantify the potential benefits of demand-side management and other power system simulation studies (e.g. home energy management, large-scale residential demand response), synthetic load datasets that accurately characterize the system load are required. This study designs a combined top-down and bottom-up approach for modelling individual residential customers and their individual electric assets, each possessing their own characteristics, using time-varying queueing models. The aggregation of all customer loads created by the queueing models represents a known city-sized load curve to be used in simulation studies. The three presented residential queueing load models use only publicly available data. An open-source Python tool to allow researchers to generate residential load data for their studies is also provided. The simulation results presented consider the ComEd region (utility company from Chicago, IL) and demonstrate the characteristics of the three proposed residential queueing load models, the impact of the choice of model parameters, and scalability performance of the Python tool.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2024

Factors Influencing Grid-connected Heat Pump Water Heater Performance in the Southeast U.S.

Grid-connected heat pump water heaters (HPWH) can shift electrical load while minimizing impacts to hot water availability for occupants. This capability provides a flexible grid resource to utilities seeking to manage peak loads Such load control also can feasibly improve renewable utilization within the utility electric production mix, for instance using off-peak generation during periods with high renewable energy generation. It also offers lower electric bills to customers through increased energy efficiency of HPWH and cuts greenhouse gas emissions. In particular, the Southeast U.S. due to its high penetration of electric water heating presents a promising opportunity for grid-connected HPWHs. This paper builds upon the results of an extensive HPWH load shifting field study conducted in 51 occupied homes in Florida using EcoPort technology (Butzbaugh et al, 2022). In 2022, only an initial evaluation was available. Here, long term load results are available as well as examination of various control strategies and influences. Analysis is conducted for HPWH energy use and load shifting performance based on home occupancy (i.e., low and high) and water heater location (i.e., conditioned and unconditioned) across different temperature profiles. An unexpected outcome of this analysis was the poor performance of HPWHs located in conditioned spaces, possibly because of inadequate air volume from improper installation. We did find higher demand reductions from 2-hour load ups and slightly improved for critical peak signals in the afternoon control periods. As expected, higher occupancy households showed great load reductions.

Fenaughty, Karen↗
DOE OSTI2020

Regenerative Fuel Cell System (Cooperative Research and Development Final Report, CRADA Number CRD-16-635)

Grid energy storage is required to maintain reliable electricity delivery from energy producers to their customers. As electrical loads on the grid change throughout the day, stored energy supplies electricity during increased power demand periods. Further, as more renewable and alternative energy sources are added, energy storage will maximize the usefulness of these technologies. Currently, the United States has over 24.6 GW of grid storage, about 2% of the total electrical production capacity. About 95% of energy storage comes from pumped hydro. However, pumped hydro energy storage is dependent on existing geographic features, making it difficult to integrate with widespread renewable energy sources. As energy demands continue to expand, and more renewable energy (i.e. wind and solar) is added to the grid, new energy storage technologies will be needed that are not dependent on geographic features. Battery technologies can provide energy storage for some applications, but are not economically well-suited for long duration charge/discharges, such as load-leveling of renewable energy. Consequently, development of new energy storage devices will augment the existing grid and reduce the capital investment in construction upgrades. As ever-increasing mandates for renewable energy are implemented, lower-cost energy storage solutions for renewable energy will be necessary to keep electricity costs low for consumers.

13 HYDRO ENERGY↗
DOE OSTI2024

Predictive model using artificial neural network to design phase change material-based ocean thermal energy harvesting systems for powering uncrewed underwater vehicles

Uncrewed Underwater Vehicles (UUVs) are a major beneficiary of the phase change material (PCM)-based ocean thermal energy harvesting technology for their mission needs. However, this technology relies on different parameters and energy conversion steps that could be critical to the general energy generation efficiency. Sea trials showed that the design performed lower than their laboratory design specifications. This underperformance results from different factors, mainly the UUV’s trajectory, travel time, underwater ocean currents, temperature fluctuations, and biofouling on the heat exchanger due to long term underwater operations. Therefore, there exists a need to continuously monitor the ambient energy harvesting system and predict system performance, for mission planning purposes. Two major parameters influencing the energy harvesting system include the final pressure inside the hydraulic energy storage vessel or accumulator, and the electrical load value. Here, this work focuses on the hydraulic to electric energy conversion system. Therefore, a combination of numerical model and experimental testing is used to develop a predictive model using artificial neural network using MATLAB. After validation with experimental testing, 1000 data samples obtained from the numerical model are used to train the ANN. Compared to the experimental results, the developed ANN model can predict in less than a second the designed benchtop system’s total efficiency with less than 15 percent maximum error range. This predictive model development represents a cost-effective way for optimization and a computational energy efficient mode aboard UUVs for mission planning for deployed UUVs using PCM-based ocean thermal energy harvesting technology.

30 DIRECT ENERGY CONVERSION↗
DOE OSTI2024

High Performance Computing Peak Shaving for Microreactor Operation

There are multiple nuclear microreactors currently under development that are designed to provide autonomous power for as many as ten or more years without refueling and are designed to power high performance computing (HPC) datacenters. But the load-follow speeds for a nuclear microreactor will be much slower than grid power and slower than the power variance typical of a HPC system. HPC datacenters experience peak power load variance driven by several factors ranging from the operation of cooling systems to remove heat from the servers to supporting a wide range of user application workflows and architectures each with different power signatures. One mechanism to support the limited load-follow of a microreactor is peak shaving where an energy storage mechanism is used to shed peak load and reduce significant power variance. This work explores peak electrical load shaving using uninterruptible power supply (UPS) systems designed for HPC support in the context of peak shaving when operating using a nuclear microreactor with a load-follow limited to 10% of load per minute. Using a self contained HPC datacenter complete with stand-alone cooling system and provisioned with an x86 cluster, an ARM cluster, and a graphics processing unit (GPU) cluster, peak shaving for microreactor operation using the UPS battery backup is explored while running two classes of typical HPC user applications. HPC architecture suitability for microreactor operation under this type of peak shaving is examined.

97 MATHEMATICS AND COMPUTING↗