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71 pages, 5996 KiB  
Review
Power Quality Impact and Its Assessment: A Review and a Survey of Lithuanian Industrial Companies
by Vladislav Liubčuk, Virginijus Radziukynas, Gediminas Kairaitis and Darius Naujokaitis
Inventions 2025, 10(2), 30; https://doi.org/10.3390/inventions10020030 (registering DOI) - 5 Apr 2025
Viewed by 31
Abstract
Poor PQ is a partial case of power system impact on society and the environment. Although the significance of good PQ is generally understood, the topic has not yet been sufficiently explored in the scientific literature. Firstly, this paper discusses the role of [...] Read more.
Poor PQ is a partial case of power system impact on society and the environment. Although the significance of good PQ is generally understood, the topic has not yet been sufficiently explored in the scientific literature. Firstly, this paper discusses the role of PQ in sustainable development by distinguishing economic, environmental, and social parts, including the existing PQ impact assessment methods. PQ problems must be studied through such prisms as financial losses of industrial companies, damage to end-use equipment, natural phenomena, interaction with animals, and social issues related to law, people’s well-being, health and safety. Secondly, this paper presents the results of the survey of Lithuanian industrial companies, which focuses on the assessment of industrial equipment immunity to both voltage sags and supply interruptions, as well as a unique methodology based on expert assessment, IEEE Std 1564-2014 and EN 50160:2010 voltage sag tables, matrix theory, a statistical hypothesis test, and convolution-based sample comparison that was developed for this purpose. The survey was carried out during the PQ monitoring campaign in the Lithuanian DSO grid, and is one of the few PQ surveys presented in the scientific literature. After counting the votes and introducing the rating system (with and without weights), the samples are compared both qualitatively and quantitatively in order to determine whether the PQ impact on various end-use equipment is similar or not. Full article
(This article belongs to the Special Issue Innovative Strategy of Protection and Control for the Grid)
15 pages, 1019 KiB  
Article
Optimal Operation of a Tablet Pressing Machine Using Deep-Neural-Network-Embedded Mixed-Integer Linear Programming
by Jialong Li, Lan Wu, Yuang Qin and Haojun Zhi
Inventions 2025, 10(2), 29; https://doi.org/10.3390/inventions10020029 - 24 Mar 2025
Viewed by 221
Abstract
This paper presents a deep neural network (DNN)-embedded mixed-integer linear programming (MILP) model for fault prediction and production optimization in tablet pressing machines. The DNN predicts the probability of failures during the tablet pressing process by analyzing key operational parameters such as pressure, [...] Read more.
This paper presents a deep neural network (DNN)-embedded mixed-integer linear programming (MILP) model for fault prediction and production optimization in tablet pressing machines. The DNN predicts the probability of failures during the tablet pressing process by analyzing key operational parameters such as pressure, temperature, humidity, speed, vibration, and number of maintenance cycles. The MILP model optimizes the temperature and humidity settings, production schedules, and maintenance planning to maximize total profit while minimizing penalties for fault pressing, energy consumption, and maintenance costs. To integrate DNN into the MILP framework, Big-M constraints are applied to linearize the Rectified Linear Unit (ReLU) activation functions, ensuring solvability and global optimality of the optimization problem. A case study using the Kaggle dataset demonstrates the model’s ability to dynamically adjust production and maintenance schedules, enhancing profitability and resource utilization under fluctuating electricity prices. Sensitivity analyses further highlight the model’s robustness to variations in maintenance and energy costs, striking an effective balance between cost efficiency and production quality, which makes it a promising solution for intelligent scheduling and optimization in complex manufacturing environments. Full article
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17 pages, 2296 KiB  
Article
Bayesian Networks Applied to the Maritime Emissions Trading System: A Tool for Decision-Making in European Ports
by Javier Vaca-Cabrero, Nicoletta González-Cancelas, Alberto Camarero-Orive and Jorge Quijada-Alarcón
Inventions 2025, 10(2), 28; https://doi.org/10.3390/inventions10020028 - 19 Mar 2025
Viewed by 174
Abstract
This study examines the impact of monitoring, reporting, and verification (MRV) system indicators on the costs associated with the emissions trading system (ETS) of the maritime sector in the European Union. Since maritime transport has recently been incorporated into the ETS, it becomes [...] Read more.
This study examines the impact of monitoring, reporting, and verification (MRV) system indicators on the costs associated with the emissions trading system (ETS) of the maritime sector in the European Union. Since maritime transport has recently been incorporated into the ETS, it becomes essential to understand how different operational and environmental factors affect the economic burden of shipping companies and port competitiveness. To this end, a model based on Bayesian networks is used to analyse the interdependencies between key variables, facilitating the identification of the most influential factors in the determination of the costs of the ETS. The results show that fuel efficiency and CO2 emissions in port are decisive in the configuration of costs. In particular, it was identified that emissions during the stay in port have a greater weight than expected, which suggests that strategies such as the use of electrical connections in port (cold ironing) may be key to mitigating costs. Likewise, navigation patterns and traffic regionalisation show a strong correlation with ETS exposure, which could lead to adjustments in maritime routes. This probabilistic model offers a valuable tool for strategic decision-making in the maritime sector, benefiting shipping companies, port operators, and policymakers. However, future research could integrate new technologies and regulatory scenarios to improve the accuracy of the analysis and anticipate changes in the ETS cost structure. Full article
(This article belongs to the Special Issue Innovations and Inventions in Ocean Energy Engineering)
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22 pages, 2171 KiB  
Article
XGBoost-Based Heuristic Path Planning Algorithm for Large Scale Air–Rail Intermodal Networks
by Shengyuan Weng, Xinghua Shan, Guangdong Bai, Jinfei Wu and Nan Zhao
Inventions 2025, 10(2), 27; https://doi.org/10.3390/inventions10020027 - 7 Mar 2025
Viewed by 377
Abstract
It is particularly important to develop efficient air–rail intermodal path planning methods for making full use of the advantages of air–rail intermodal networks and providing passengers with richer and more reasonable travel options. A Time-Expanded Graph (TEG) is used to model the timetable [...] Read more.
It is particularly important to develop efficient air–rail intermodal path planning methods for making full use of the advantages of air–rail intermodal networks and providing passengers with richer and more reasonable travel options. A Time-Expanded Graph (TEG) is used to model the timetable information of public transportation providing a theoretical basis for public transportation path planning. However, if the TEG includes a large amount of data such as train stations, airports, train and air schedules, the network scale will become very large, making path planning extremely time-consuming. This study proposes an XGBoost-based heuristic path planning algorithm (XGB-HPPA) for large scale air–rail intermodal networks, which use the XGBoost model to predict transfer stations before path planning, and quickly eliminate unreasonable transfer edges by adding a heuristic factor, reducing the network scale, thus accelerating the computation speed. Comparative results indicate that XGB-HPPA can markedly enhance computational speed within large-scale networks, while obtaining as many valid solutions as possible and approximating the optimal solution. Full article
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12 pages, 1604 KiB  
Article
CrySPAI: A New Crystal Structure Prediction Software Based on Artificial Intelligence
by Zongguo Wang, Ziyi Chen, Yang Yuan and Yangang Wang
Inventions 2025, 10(2), 26; https://doi.org/10.3390/inventions10020026 - 6 Mar 2025
Viewed by 409
Abstract
Crystal structure predictions based on the combination of first-principles calculations and machine learning have achieved significant success in materials science. However, most of these approaches are limited to predicting specific systems, which hinders their application to unknown or unexplored domains. In this paper, [...] Read more.
Crystal structure predictions based on the combination of first-principles calculations and machine learning have achieved significant success in materials science. However, most of these approaches are limited to predicting specific systems, which hinders their application to unknown or unexplored domains. In this paper, we present a crystal structure prediction software based on artificial intelligence, named as CrySPAI, to predict energetically stable crystal structures of inorganic materials given their chemical compositions. The software consists of three key modules, an evolutionary optimization algorithm (EOA) that searches for all possible crystal structure configurations, density functional theory (DFT) that provides the accurate energy values for these structures, and a deep neural network (DNN) that learns the relationship between crystal structures and their corresponding energies. To optimize the process across these modules, a distributed framework is implemented to parallelize tasks, and an automated workflow has been integrated into CrySPAI for seamless execution. This paper reports the development and implementation of the AI-based CrySPAI Crystal Prediction Software tool and its unique features. Full article
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15 pages, 4727 KiB  
Article
Research on Partial Discharge Spectrum Recognition Technology Used in Power Cables Based on Convolutional Neural Networks
by Zhenqing Zhang, Hao Wu, Weiyin Ren, Jian Yan, Zhefu Sun and Man Ding
Inventions 2025, 10(2), 25; https://doi.org/10.3390/inventions10020025 - 5 Mar 2025
Viewed by 427
Abstract
Partial discharge is an important symptom of cable aging, and timely detection of potential defects is of great significance to ensure the stability and safety of the power supply. However, due to the diversity of inspection equipment and information blockage, the staff often [...] Read more.
Partial discharge is an important symptom of cable aging, and timely detection of potential defects is of great significance to ensure the stability and safety of the power supply. However, due to the diversity of inspection equipment and information blockage, the staff often show blindness to the partial discharge spectrum and the defects corresponding to the spectrum. In view of this phenomenon, a partial discharge spectrum recognition method based on a convolutional neural network was developed. Firstly, a database of typical partial discharge spectrum was established, including partial amplifiers in the laboratory and at the work site, and then the convolutional neural network was used to train the defect spectral library. This paper proposes a processing technology for the on-site partial discharge spectrum; the unified grayscale image is obtained by grayscale processing, linearized stretching and size unification, and then the shape and color feature parameters are extracted according to the grayscale image, which solves the image distortion and statistical spectrum movement caused by the on-site environment or photographic angle on the user side. The partial discharge type can be obtained by comparing the processed spectrum with the database through the intelligent terminal, which greatly improves the accuracy and efficiency of on-site operations. Full article
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25 pages, 3293 KiB  
Article
Design of a Semi-Continuous Microwave System for Pretreatment of Microwave-Assisted Pyrolysis Using a Theoretical Method
by Paula Andrea Ramírez Cabrera, Alejandra Sophia Lozano Pérez and Carlos Alberto Guerrero Fajardo
Inventions 2025, 10(2), 24; https://doi.org/10.3390/inventions10020024 - 4 Mar 2025
Viewed by 606
Abstract
This article provides an overview of various microwave-assisted techniques, such as microwave-assisted extraction (MAE), microwave-assisted organic synthesis (MAOS), microwave-assisted pyrolysis (MAP), microwave-assisted hydrothermal treatment (MAHT), microwave-assisted acid hydrolysis (MAAH), microwave-assisted organosolv (MAO), microwave-assisted alkaline hydrolysis (MAA), microwave-assisted enzymatic hydrolysis (MAEH), and microwave-assisted fermentation [...] Read more.
This article provides an overview of various microwave-assisted techniques, such as microwave-assisted extraction (MAE), microwave-assisted organic synthesis (MAOS), microwave-assisted pyrolysis (MAP), microwave-assisted hydrothermal treatment (MAHT), microwave-assisted acid hydrolysis (MAAH), microwave-assisted organosolv (MAO), microwave-assisted alkaline hydrolysis (MAA), microwave-assisted enzymatic hydrolysis (MAEH), and microwave-assisted fermentation (MAF). Microwave-assisted biomass pretreatment has emerged as a promising method to improve the efficiency of biomass conversion processes, in particular microwave-assisted pyrolysis (MAP). The focus is on microwave-assisted pyrolysis, detailing its key components, including microwave sources, applicators, feedstock characteristics, absorbers, collection systems, and reactor designs. Based on different studies reported in the literature and a mathematical model, a mechanical design of a microwave oven adapted for pyrolysis is proposed together with a computer-aided design and a finite element analysis. The semi-continuous system is designed for a 40 L capacity and a power of 800 W. The material with which the vessel was designed is suitable for the proposed process. The challenges, opportunities, and future directions of microwave-assisted technologies for the sustainable use of biomass resources are presented. Full article
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24 pages, 6291 KiB  
Article
Internet of Things Smart Beehive Network: Homogeneous Data, Modeling, and Forecasting the Honey Robbing Phenomenon
by Igor Kurdin and Aleksandra Kurdina
Inventions 2025, 10(2), 23; https://doi.org/10.3390/inventions10020023 - 3 Mar 2025
Viewed by 745
Abstract
The role of experimental data and the use of IoT-based monitoring systems are gaining broader significance in research on bees across several aspects: bees as global pollinators, as biosensors, and as examples of swarm intelligence. This increases the demands on monitoring systems to [...] Read more.
The role of experimental data and the use of IoT-based monitoring systems are gaining broader significance in research on bees across several aspects: bees as global pollinators, as biosensors, and as examples of swarm intelligence. This increases the demands on monitoring systems to obtain homogeneous, continuous, and standardized experimental data, which can be used for machine learning, enabling models to be trained on new online data. However, the continuous operation of monitoring systems introduces new risks, particularly the cumulative impact of electromagnetic radiation on bees and their behavior. This highlights the need to balance IoT energy consumption, functionality, and continuous monitoring. We present a novel IoT-based bee monitoring system architecture that has been operating continuously for several years, using solar energy only. The negative impact of IoT electromagnetic fields is minimized, while ensuring homogeneous and continuous data collection. We obtained experimental data on the adverse phenomenon of honey robbing, which involves elements of swarm intelligence. We demonstrate how this phenomenon can be predicted and illustrate the interactions between bee colonies and the influence of solar radiation. The use of criteria for detecting honey robbing will help to reduce the spread of diseases and positively contribute to the sustainable development of precision beekeeping. Full article
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24 pages, 3105 KiB  
Article
Development of OptiCon: A Mathematical Model with a Graphical User Interface for Designing Sustainable Portland Cement Concrete Mixes with Budget Constraint
by Angie Pineda, Rita Peñabaena-Niebles, Gilberto Martínez-Arguelles and Rodrigo Polo-Mendoza
Inventions 2025, 10(2), 22; https://doi.org/10.3390/inventions10020022 - 1 Mar 2025
Viewed by 790
Abstract
The production of Portland Cement Concrete (PCC) generates significant environmental impacts that increase climate change and decrease people’s quality of life. Recent studies highlight the potential to reduce these environmental burdens by partially replacing Portland cement with Supplementary Cementitious Materials (SCMs) and coarse [...] Read more.
The production of Portland Cement Concrete (PCC) generates significant environmental impacts that increase climate change and decrease people’s quality of life. Recent studies highlight the potential to reduce these environmental burdens by partially replacing Portland cement with Supplementary Cementitious Materials (SCMs) and coarse aggregates with Recycled Concrete Aggregate (RCA). However, designing PCCs with simultaneous contents of SCMs and RCA is not easily manageable because current design procedures fail to adjust all of the variables involved. In order to overcome these limitations, this research introduces a novel mathematical model designed to develop operationally efficient PCC mixes that are both environmentally sustainable and cost-effective. The proposed model, denominated OptiCon, employs the Life-Cycle Assessment and Life-Cycle Costs Analysis methodologies to evaluate the incorporation of three different SCMs (i.e., fly ash, silica fume, and steel slag) and RCA into PCC mixes. OptiCon is also integrated within a graphical user interface in order to make its implementation straightforward for potential users. Thus, OptiCon is operationalized through an algorithm, offering a replicable approach that can be adapted to various contexts, providing both a theoretical framework and a practical tool for state agencies, engineers, suppliers, and other stakeholders to adopt more environmentally friendly practices in concrete production. Furthermore, a case study from northern Colombia analyzed thirty mix design scenarios with varying supplier conditions (foreign, local, or mixed), calculating costs and CO2 emissions for a fixed concrete volume of 1 m3. The findings demonstrated that utilizing OptiCon can achieve substantial reductions in both CO2 emissions and production costs, underscoring the model’s efficiency and practical impact. Full article
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11 pages, 2793 KiB  
Article
Automatic Phase Correction of NMR Spectra Using Brute-Force GPU Method
by Mario Gazziro, Marcio Luís Munhoz Amorim, Marco Roberto Cavallari, João Paulo Carmo, Alberto Tannus, Oswaldo Hideo Ando Junior and Loren Schwiebert
Inventions 2025, 10(2), 21; https://doi.org/10.3390/inventions10020021 - 1 Mar 2025
Viewed by 593
Abstract
Although there are still no fully guaranteed solutions to the problem of phase adjustment of NMR spectroscopy signals, it has not received much consideration recently, especially in the presence of noisy signals. To address this gap, we present a novel methodology, based on [...] Read more.
Although there are still no fully guaranteed solutions to the problem of phase adjustment of NMR spectroscopy signals, it has not received much consideration recently, especially in the presence of noisy signals. To address this gap, we present a novel methodology, based on GPU processing, that is able to find the optimal parameter set for phase adjustment through an exhaustive search of all possible combinations of the phase space parameters. In our experiments, we were able to reduce the execution time of extensive GPU brute-force analysis to the same amount of time needed for the traditional CPU analysis, with the big advantage of searching all possible combinations on the GPU against just a few regions guessed by the CPU. In our case study, we also demonstrate the robustness of the proposed method with respect to the problem of local minima. Finally, we perform a Bland-Altman analysis to validate the entropies calculated using CPU and GPU processing for a set of 16 experiments from brain and body metabolites using 1H and 31P probes. The results demonstrate that our algorithm always find the globally optimal solution while previous CPU-based heuristics were stalled in a poor solution in 6.25% of a 16 sample universe. Full article
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37 pages, 75683 KiB  
Article
Development and Usability Evaluation of Inventive and Repurposable Children’s Furniture
by Zhi Yuan Phuah, Poh Kiat Ng, Chai Hua Tay, Boon Kian Lim, Kia Wai Liew and Peng Lean Chong
Inventions 2025, 10(1), 20; https://doi.org/10.3390/inventions10010020 - 17 Feb 2025
Viewed by 709
Abstract
This study developed inventive and repurposable children’s furniture to improve functionality and extend product lifespan. Unlike typical cribs which serve a single purpose, the proposed design supports multiple functions: crib, cushioned chair, highchair, walker, toilet attachment, pull-up bar, and bed safety rail. Specific [...] Read more.
This study developed inventive and repurposable children’s furniture to improve functionality and extend product lifespan. Unlike typical cribs which serve a single purpose, the proposed design supports multiple functions: crib, cushioned chair, highchair, walker, toilet attachment, pull-up bar, and bed safety rail. Specific dimensions were established, and the correct material selections were made for the selected concept. This was finalised using Autodesk Inventor 2019 that was used for stress analysis, and material optimisation. Usability tests were conducted to compare the proposed invention with single-function products. These tests were discussed in terms of task completion time, space-saving ability, survey feedback, and a REBA of musculoskeletal risk. It was found that the proposed design could be repurposed in a shorter time and save more space. A majority of the survey participants agreed that it performed well in terms of repurposability, design, space-saving ability, usability, comfort, sustainability, and safety. Additionally, the proposed design is cheaper compared to single-function products. Thus, creating inventive and repurposable children’s furniture can contribute to reducing waste and extending the lifespan of children’s furniture through innovative, cost-effective design solutions. Full article
(This article belongs to the Section Inventions and Innovation in Advanced Manufacturing)
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19 pages, 6308 KiB  
Review
Power-Hardware-in-the-Loop Simulation for Applied Science, a Review to Highlight Its Merits and Challenges
by Ciro Núñez-Gutiérrez
Inventions 2025, 10(1), 19; https://doi.org/10.3390/inventions10010019 - 12 Feb 2025
Viewed by 606
Abstract
The last thirty years have brought an evolution of electrical power systems. The integration of renewable energy and energy storage, dynamic loads, or distributed resources based on power electronics, including communications systems and protocols, is usual. Fortunately, technological advances have also brought tools [...] Read more.
The last thirty years have brought an evolution of electrical power systems. The integration of renewable energy and energy storage, dynamic loads, or distributed resources based on power electronics, including communications systems and protocols, is usual. Fortunately, technological advances have also brought tools to face the complex field of electrical component integration to the power system, such as the real-time power-hardware-in-the-loop (PHIL) simulation. This paper argues why PHIL simulation is a mighty tool for addressing intelligent design, modeling, and computing methods to address complex power systems. Nevertheless, any promising technology can be misunderstood, reducing its positive effect. This article uses two inverters connected to a microgrid to develop the steps from conceptualizing an idea to a PHIL simulation, highlighting the merits, drawbacks, and lessons learned. Two perspectives are developed. First, the multiple, even complex, details required for furnishing a PHIL simulation are described. Second, reflections on how PHIL simulations enhance the scientific impact of the research compared to offline simulations or scale prototypes are made, enabling the transition from academic to applied research to attend to the challenges of modern power systems. The successful results of the microgrid PHIL simulation are shown to prove the merits of this approach. Full article
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12 pages, 6221 KiB  
Article
An Efficient Contamination-Reducing Closet for Reusing Protective Clothing
by Xing Qiu, Jeffery C. C. Lo, Yuanjie Cheng, Hua Xu, Qianwen Xu and Shi-Wei Ricky Lee
Inventions 2025, 10(1), 18; https://doi.org/10.3390/inventions10010018 - 10 Feb 2025
Viewed by 496
Abstract
A professional closet with highly efficient disinfection for reusing protective clothing is required to reduce supply and demand and protect the environment. A self-developed ultraviolet-C (UVC) light-emitting diode (LED) package that can emit uniform radiance in a certain distance was developed; and a [...] Read more.
A professional closet with highly efficient disinfection for reusing protective clothing is required to reduce supply and demand and protect the environment. A self-developed ultraviolet-C (UVC) light-emitting diode (LED) package that can emit uniform radiance in a certain distance was developed; and a series of disinfection modules with UVC LED packages were installed in a closet for disinfection. A disinfection module can achieve an over 99.9% disinfection rate of H1N1; E. coli; S. aureus; Pseudomonas aeruginosa; and an over 99% disinfection rate of EV71 within a minute. A 1-min disinfection closet was developed to reuse protective clothing. The closet was well-designed; as well as a series of burn-in tests were performed after the assembly of the closet. The optical and thermal properties of the closet were stable within one minute of a working period during the burn-in test. After disinfection; bacterial filtration efficiency (BFE) and viral filtration efficiency (VFE) were examined on the disposable protective clothing. The disposable protective clothing did not show any degradation after being exposed to UVC for sixty minutes; which means the defensive capability of medical protective clothing can be reused sixty times in light of the self-developed disinfection closet. The disinfection closet provides an efficient method for reusing protective clothing. Full article
(This article belongs to the Section Inventions and Innovation in Advanced Manufacturing)
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11 pages, 3760 KiB  
Article
Analysis of Contact Noise Due to Elastic Recovery of Surface Asperities for Spherical Contact
by Bora Lee, Kyungseob Kim and Taewan Kim
Inventions 2025, 10(1), 17; https://doi.org/10.3390/inventions10010017 - 8 Feb 2025
Viewed by 501
Abstract
Contact noise, often arising from frictional vibrations in mechanical systems, significantly impacts performance and user experience. This study investigates the generation of contact noise due to the elastic recovery of surface asperities during spherical contact with rough surfaces. A numerical algorithm was developed [...] Read more.
Contact noise, often arising from frictional vibrations in mechanical systems, significantly impacts performance and user experience. This study investigates the generation of contact noise due to the elastic recovery of surface asperities during spherical contact with rough surfaces. A numerical algorithm was developed to model the noise produced by the elastic–plastic deformation of asperities, incorporating surface roughness and normal load effects. Gaussian-distributed rough surfaces with varying Ra values (0.01–5 μm) were generated to analyze the interaction between a rigid sphere and the rough surface. Contact pressure, asperity deformation, and the resulting acoustic emissions were calculated. The results indicate that, as surface roughness and applied load increase, noise levels within the audible frequency range also rise, exceeding 70 dB under certain conditions. The transition from elastic to plastic deformation significantly influences the noise characteristics. Surfaces with Ra ≥ 0.1 μm showed a 10–15 dB increase in noise compared to smoother surfaces. These findings offer insights into optimizing surface parameters for noise reduction in rolling contact applications, providing a foundation for designing low-noise mechanical systems. Future experimental validations are expected to enhance the practical applications of this analytical framework. Full article
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16 pages, 491 KiB  
Article
A Stackelberg Game Model for the Energy–Carbon Co-Optimization of Multiple Virtual Power Plants
by Dayong Xu and Mengjie Li
Inventions 2025, 10(1), 16; https://doi.org/10.3390/inventions10010016 - 8 Feb 2025
Viewed by 513
Abstract
As energy and carbon markets evolve, it has emerged as a prevalent trend for multiple virtual power plants (VPPs) to engage in market trading through coordinated operation. Given that these VPPs belong to diverse stakeholders, a competitive dynamic is shaping up. To strike [...] Read more.
As energy and carbon markets evolve, it has emerged as a prevalent trend for multiple virtual power plants (VPPs) to engage in market trading through coordinated operation. Given that these VPPs belong to diverse stakeholders, a competitive dynamic is shaping up. To strike a balance between the interests of the distribution system operator (DSO) and VPPs, this paper introduces a bi-level energy–carbon coordination model based on the Stackelberg game framework, which consists of an upper-level optimal pricing model for the DSO and a lower-level optimal energy scheduling model for each VPP. Subsequently, the Karush-Kuhn-Tucker (KKT) conditions and the duality theorem of linear programming are applied to transform the bi-level Stackelberg game model into a mixed-integer linear program, allowing for the computation of the model’s global optimal solution using commercial solvers. Finally, a case study is conducted to demonstrate the effectiveness of the proposed model. The simulation results show that the proposed game model effectively optimizes energy and carbon pricing, encourages the active participation of VPPs in electricity and carbon allowance sharing, increases the profitability of DSOs, and reduces the operational costs of VPPs. Full article
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