Search Results (1,543)

As energy systems transition toward high shares of variable renewable generation, local energy communities (ECs) are increasingly relevant for enabling demand-side flexibility and self-sufficiency. This shift is particularly evident in the residential sector, where the deployment of photovoltaic (PV) systems is rapidly growing. While mixed-integer linear programming (MILP) remains the standard for operational optimization and demand response in such systems, its computational burden limits scalability and responsiveness under real-time or uncertain conditions. Reinforcement learning (RL), by contrast, offers a model-free, adaptive alternative. However, its application to real-world energy system operation remains limited. This study explores the application of a Deep Q-Network (DQN) to a real residential EC, which has received limited attention in prior work. The system comprises three single-family homes sharing a centralized heating system with a thermal energy storage (TES), a PV installation, and a grid connection. We compare the performance of MILP and RL controllers across economic and environmental metrics. Relative to a reference scenario without TES, MILP and RL reduce energy costs by 10.06% and 8.78%, respectively, and both approaches yield lower total energy consumption and CO${}_2$-equivalent emissions. Notably, the trained RL agent achieves a near-optimal outcome while requiring only 22% of the MILP’s computation time. These results demonstrate that DQNs can offer a computationally efficient and practically viable alternative to MILP for real-time control in residential energy systems. Full article

The soft-reset process, or a sequence of charge emissions from a floating storage node through a transistor biased in a subthreshold bias condition, is modeled by a master (Kolmogorov–Bateman) equation. The Coulomb interaction energy after each one-charge emission leads to a stepwise potential increase, giving correlated emission rates represented by Boltzmann factors. The governing probability distribution function is a hypoexponential type, and its cumulants describe characteristics of the single-charge Coulomb interaction at room temperature on a mesoscopic scale. The cumulants are further extended into a complex domain. Starting from three fundamental assumptions, i.e., the generation of non-degenerated states due to single-charge Coulomb energy, the Markovian property of each emission event, and the independence of each state, a moment function is identified as a product of mutually prime elements (algebraically termed as prime ideals) comprising the eigenvalues or the lifetimes of the emission states. Then, the algebraic structure of the moment function is found to be highly analogous to that of an integer uniquely factored into prime numbers. Treating the lifetimes as analogs of the prime numbers, two types of zeta functions are constructed. Standard analyses of the zeta functions analogous to the prime number problem or the Riemann Hypothesis are performed. For the zeta functions, the analyticity and poles are specified, and the functional equations are derived. Also, the zeta functions are found to be equivalent to the analytic extension of the cumulants. Finally, between the number of emitted charges and the lifetime, a logarithmic relation analogous to the prime number theorem is derived. Full article

Extreme damage poses a serious challenge to the safe operation of power grids. Optimizing the allocation of defense resources to improve the grid’s disaster resistance capabilities is the main concern of the power system. In this paper, a distributed robust optimal defense resource allocation method based on the defender–attacker–defender model is proposed to improve the disaster resilience of power grids. This method takes into account the uncertainty of restoration time due to different damage intensities and improves the efficiency of restoration resource scheduling in the restoration process. Meanwhile, a set covering-column and constraint generation (SC-C&CG) algorithm is proposed for the case that the mixed integer model does not satisfy the Karush–Kuhn–Tucker (KKT) condition. A case study based on the IEEE 24-bus system is conducted, and the results verify that the proposed method can minimize the system dumping load under the uncertainty of the maintenance time involved. Full article

The Minimum Spanning Tree with Conflicting Edge Pairs is a generalization that adds conflict constraints to a classical optimization problem on graphs used to model several real-world applications. In recent years, several heuristic and exact approaches have been proposed to tackle this problem. In this paper, we present a mixed-integer linear program not previously applied to this problem, and we solve it with an open-source solver. Computational results for the benchmark instances commonly adopted in the literature of the problem are reported. The results indicate that the approach we propose obtains results aligned with those of the much more sophisticated approaches available, notwithstanding it being much simpler to implement. During the experimental campaign, six instances were closed for the first time, with nine improved best-known lower bounds and sixteen improved best-known upper bounds over a total of two hundred thirty instances considered. Full article

Due to the high cost of hydrogen utilization and the uncertainties in renewable energy generation and load demand, significant challenges are posed for the operation optimization of hydrogen-containing integrated energy systems (IESs). In this study, a robust operational model for an electric–heat–gas IES (EHG-IES) is proposed, considering the hydrogen energy system heat recovery (HESHR) and multiple uncertainties. Firstly, a heat recovery model for the hydrogen system is established based on thermodynamic equations and reaction principles; secondly, through the constructed adjustable robust optimization (ARO) model, the optimal solution of the system under the worst-case scenario is obtained; lastly, the original problem is decomposed based on the column and constraint generation method and strong duality theory, resulting in the formulation of a master problem and subproblem with mixed-integer linear characteristics. These problems are solved through alternating iterations, ultimately obtaining the corresponding optimal scheduling scheme. The simulation results demonstrate that our model and method can effectively reduce the operation and maintenance costs of HESHR-EHG-IES while being resilient to uncertainties on both the supply and demand sides. In summary, this study provides a novel approach for the diversified utilization and flexible operation of energy in HESHR-EHG-IES, contributing to the safe, controllable, and economically efficient development of the energy market. It holds significant value for engineering practice. Full article

This work presents the design and implementation of an automated, digital, and modular system to address a real-world industrial challenge: the automation and optimization of production schedules for Computer Numerical Control (CNC) machines in a factory in Portugal. The goal is to replicate and enhance the existing manual scheduling process by integrating multiple data sources and formulating a general Mixed-Integer Linear Programming (MILP) model with constraints. This model can be solved using MILP optimization methods to produce efficient scheduling solutions that minimize machine downtime, reduce tool change frequency, and lower operator workload. The proposed system is implemented using open-source Python abstraction interfaces (Python-MIP), employing state-of-the-art of MILP optimization solvers such as CBC and HiGHS for solution validation. The system is designed to accommodate a wide range of constraints and operational factors, which can be switched on or off as needed, thereby enhancing its flexibility and decision-support capabilities. Additionally, a user-friendly graphical application is developed to facilitate the input of specific scheduling data and constraints, enabling flexible and efficient formulation of diverse scheduling scenarios. The proposed system is validated through multiple case studies, demonstrating its effectiveness in optimizing industrial CNC scheduling tasks and providing a scalable, practical tool for real-world factory operations. Full article

We present a formal reconstruction of the conventional number systems, including integers, rationals, reals, and complex numbers, based on the principle of relational finitude over a finite field ${\mathbb{F}}_{\mathsf{p}}$. Rather than assuming actual infinity, we define arithmetic and algebra as observer-dependent constructs grounded in finite field symmetries. Consequently, we formulate relational analogues of the conventional number classes, expressed relationally with respect to a chosen reference frame. We define explicit mappings for each number class, preserving their algebraic and computational properties while eliminating ontological dependence on infinite structures. For example, relationally framed rational numbers emerge from dense grids generated by primitive roots of a finite field, enabling proportional reasoning without infinity, while scale-periodicity ensures invariance under zoom operations, approximating continuity in a bounded structure. The resultant framework—that we denote as Finite Ring Continuum—aims to establish a coherent foundation for mathematics, physics and formal logic in an ontologically finite paradox-free informational universe. Full article

Vehicle electrification is one of the main development directions within the automobile industry. However, due to the range limit of electric vehicles, electric vehicle users generally have range anxiety, especially toward long-haul driving. Therefore, there is an urgent need to effectively coordinate route planning and charging during long-haul driving, especially considering factors such as insufficient charging facilities, long charging times, battery aging, and changes in energy consumption under variable-temperature environments. In this study, the goal is to collaboratively optimize route planning and charging strategies. To achieve this goal, a mixed-integer nonlinear model is developed to minimize the total system cost, an electrochemical model is applied to accurately track the battery state, and a two-layer IACO-SA is proposed. Finally, the highway network in five provinces of China is adopted as an example to compare the optimal scheme results of our model with those of three other models. The comparison results prove the effectiveness of the proposed model and solution algorithm for the collaborative optimization of route planning and charging strategies of electric vehicles during long-haul driving. Full article

This article introduces a flexible time series regression model known as the Mixture of Integer-Valued Generalized Autoregressive Conditional Heteroscedasticity (MINGARCH). Mixture models provide versatile frameworks for capturing heterogeneity in count data, including features such as multiple peaks, seasonality, and intervention effects. The proposed model is applied to regional COVID-19 data from Malaysia. To account for geographical variability, five regions—Selangor, Kuala Lumpur, Penang, Johor, and Sarawak—were selected for analysis, covering a total of 86 weeks of data. Comparative analysis with existing time series regression models demonstrates that MINGARCH outperforms alternative approaches. Further investigation into forecasting reveals that MINGARCH yields superior performance in regions with high population density, and significant influencing factors have been identified. In low-density regions, confirmed cases peaked within three weeks, whereas high-density regions exhibited a monthly seasonal pattern. Forecasting metrics—including MAPE, MAE, and RMSE—are significantly lower for the MINGARCH model compared to other models. These results suggest that MINGARCH is well-suited for forecasting disease spread in urban and densely populated areas, offering valuable insights for policymaking. Full article

The widespread deployment of electric vehicles (EVs) has introduced substantial challenges to electricity pricing, grid stability, and renewable energy integration. This paper proposes a real-time pricing optimization framework for large-scale EV charging networks incorporating renewable intermittency, demand elasticity, and infrastructure constraints within a high-dimensional optimization model. The core objective is to dynamically determine spatiotemporal electricity prices that simultaneously reduce system peak load, improve renewable energy utilization, and minimize user charging costs. A rigorous mathematical formulation is developed integrating over 40 system-level constraints, including power balance, transmission capacity, renewable curtailment, carbon targets, voltage regulation, demand-side flexibility, social participation, and cyber resilience. Real-time electricity prices are treated as dynamic decision variables influenced by charging station utilization, elasticity response curves, and the marginal cost of renewable and grid-supplied electricity. The problem is solved over 96 time intervals using a hybrid solution approach, with benchmark comparisons against mixed-integer programming (MILP) and deep reinforcement learning (DRL)-based baselines. A comprehensive case study is conducted on a 500-station EV charging network serving 10,000 vehicles integrated with a modified IEEE 118-bus grid model and 800 MW of variable renewable energy. Historical charging data with ±12% stochastic demand variation and real-world solar and wind profiles are used to simulate realistic operational conditions. Results demonstrate that the proposed framework achieves a 23.4% average peak load reduction per station, a 17.9% improvement in renewable energy utilization, and user cost savings of up to 30% compared to baseline flat-rate pricing. Utilization imbalances across the network are reduced, with congestion mitigation observed at over 90% of high-traffic stations. The real-time pricing model successfully aligns low-price windows with high-renewable periods and off-peak hours, achieving time-synchronized load shifting and system-wide flexibility. Visual analytics including high-resolution 3D surface plots and disaggregated bar charts reveal structured patterns in demand–price interactions, confirming the model’s ability to generate smooth, non-disruptive pricing trajectories. The results underscore the viability of advanced optimization-based pricing strategies for scalable, clean, and responsive EV charging infrastructure management in renewable-rich grid environments. Full article

In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the one-dimensional curve bounding a disk. For any non-negative integer, a circle is called n-enclosing if it contains exactly n lattice points on the $xy$-plane in its interior. In this paper, we are mainly interested in when the largest n-enclosing circle exists and what the largest radius is. We study the small integer cases by hand and extend to all $n<1100$ with the aid of a computer. We find that frequently such a circle does not exist, e.g., when $n=5,6$. We then show a few general results on these circles including some regularities among their radii and an easy criterion to determine exactly when the largest n-enclosing circles exist. Further, from numerical evidence, we conjecture that the set of integers whose largest enclosing circles exist is infinite, and so is its complementary in the set of non-negative integers. Throughout this paper, we present more mysteries/problems/conjectures than answers/solutions/theorems. In particular, we list many conjectures and some unsolved problems including possible higher-dimensional generalizations at the end of the last two sections. Full article

The current paper presents a novel numerical technique to handle variable-order multiterm fractional differential equations (VO-MTFDEs) supplemented with initial conditions (ICs) by introducing generalized fractional Jacobi functions (GFJFs). These GFJFs satisfy the associated ICs. A crucial part of this approach is using the spectral collocation method (SCM) and building operational matrices (OMs) for both integer-order and variable-order fractional derivatives in the context of GFJFs. These lead to efficient and accurate computations. The suggested algorithm’s convergence and error analysis are proved. The feasibility of the suggested procedure is confirmed via five numerical test examples. Full article

With the rapid growth of renewable energy integration, power systems are facing increasing uncertainty and variability in operation. The intermittent and uncontrollable nature of wind and solar generation requires operational decisions to anticipate future fluctuations, creating strong temporal coupling across days. This leads to large-scale mixed-integer linear programming (MILP) with a large number of binary variables, which is computationally intensive—especially in year-long simulations. As a result, there is a growing need for efficient modeling approaches that can reduce complexity while preserving key temporal features. This paper proposes a temporal–spatial acceleration framework for long-term power system operation simulation. In the temporal dimension, a monthly K-means clustering algorithm is applied to reconstruct typical scenario days from 8760 h time series, preserving the characteristics of seasonal and intraday variability. In the spatial dimension, thermal units with similar characteristics are aggregated, and binary decision variables are relaxed into continuous variables, transforming the MILP into a tractable LP model, and thereby reducing computational burden. Case studies are performed based on the six-bus and the IEEE RTS-79 systems to validate the framework, being able to provide a practical solution for renewable-integrated power system planning and dispatch applications. Full article

To promote a more decentralized energy system, the European Commission introduced the concept of Renewable Energy Communities (RECs). Meanwhile, the increasing penetration of Electric Vehicles (EVs) may significantly increase peak power demand and consumption ramps when charging sessions are left uncontrolled. However, by integrating smart charging strategies, such as Vehicle-to-Grid (V2G), EV storage can actively support the energy balance within RECs. In this context, this work proposes a comprehensive and scalable model for leveraging smart charging capabilities in RECs. This approach focuses on an external cooperative framework to optimize incentive acquisition and reduce dependence on Medium Voltage (MV) grid substations. It adopts a hybrid strategy, combining Mixed-Integer Linear Programming (MILP) to solve the day-ahead global optimization problem with local rule-based controllers to manage power deviations. Simulation results for a six-month case study, using historical demand data and synthetic charging sessions generated from real-world events, demonstrate that V2G integration leads to a better alignment of overall power consumption with zonal pricing, smoother load curves with a 15.5% reduction in consumption ramps, and enhanced cooperation with a 90% increase in shared power redistributed inside the REC. Full article

Driven by the need for sustainable and efficient energy systems, the optimal management of distributed generation, including photovoltaic systems and battery energy storage systems within prosumer households, is of crucial importance. This requires a comprehensive cost–benefit analysis to assess their viability. In this study, an optimization model formulated as a mixed-integer linear programming problem is proposed to evaluate the integration of battery storage systems for 10 prosumers on the radial feeder in Croatia and to quantify the benefits both from the prosumers’ perspective and that of the reduction in grid losses. The results show significant annual cost reductions for prosumers, totaling EUR 1798.78 for the observed feeder, with some achieving a net profit. Grid losses are significantly reduced by 1172.52 kWh, resulting in an annual saving of EUR 216.25 for the distribution system operator. However, under the current Croatian market conditions, the integration of battery storage systems is not profitable over the entire lifetime due to the high initial investment costs of EUR 720/kWh. The break-even analysis reveals that investment cost needs to decrease by 52.78%, or an inflation rate of 4.87% is required, to reach prosumer profitability. This highlights the current financial barriers to the widespread adoption of battery storage systems and emphasizes the need for significant cost reductions or targeted incentives. Full article