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18 pages, 898 KB

Open AccessArticle

Methodical Aspects of Calculation of Technical Energy Losses in a Direct Current Electric Network

by Alexey Kirpikov, Vladislav Oboskalov, Murodbek Safaraliev, Ismoil Odinaev, Mihail Senyuk and Svetlana Beryozkina

Mathematics 2026, 14(12), 2228; https://doi.org/10.3390/math14122228 - 22 Jun 2026

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Abstract

This paper addresses probabilistic and statistical methods for calculating technical energy losses in direct current (DC) networks. A DC network model is adopted as the basis for the analysis, and several approaches are compared in terms of qualitative features and computational efficiency. The [...] Read more.

This paper addresses probabilistic and statistical methods for calculating technical energy losses in direct current (DC) networks. A DC network model is adopted as the basis for the analysis, and several approaches are compared in terms of qualitative features and computational efficiency. The load profile is described using probabilistic indicators, emphasizing the importance of accounting for correlation moments (CMs) between node powers and CMs between voltages to reduce calculation errors. A correction procedure for the mathematical expectation of node voltages is proposed, which significantly improves the accuracy of loss estimation. Simulation studies on representative four-node DC test networks show that the proposed method reduces the root mean square error in loss estimation by up to 15–20% compared with traditional approaches based solely on mean load values. The results confirm that the correction of node voltage expectations provides a good balance between accuracy and computational cost and can be recommended as an independent procedure within existing probabilistic frameworks for loss assessment. Full article

(This article belongs to the Special Issue Mathematical Applications in Electrical Engineering, 2nd Edition)

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14 pages, 1538 KB

Open AccessArticle

A Very Fast Method for Solving Electrical Circuits with Nonlinear Resistive Elements

by Claudiu Tufan, Alexandru Gabriel Gheorghe and George Marian Vasilescu

Mathematics 2026, 14(9), 1493; https://doi.org/10.3390/math14091493 - 29 Apr 2026

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Abstract

We present an improved version of the Hănțilă Method (HM) for circuits with multiple nonlinear resistive elements (NRCEs). The method replaces NRCEs with generators consisting of linear resistors and nonlinear controlled sources, whose values are updated iteratively. At each iteration, the resulting linear [...] Read more.

We present an improved version of the Hănțilă Method (HM) for circuits with multiple nonlinear resistive elements (NRCEs). The method replaces NRCEs with generators consisting of linear resistors and nonlinear controlled sources, whose values are updated iteratively. At each iteration, the resulting linear circuit is solved in the frequency domain for each harmonic. We evaluate the computational effort and compare computation times and results with those obtained using conventional methods applied to a circuit comprising parallel-connected single-phase half-wave rectifiers. Full article

(This article belongs to the Special Issue Mathematical Applications in Electrical Engineering, 2nd Edition)

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22 pages, 879 KB

Open AccessArticle

Formal Integration of ISO/IEC Digital Twin Standards: A Layered Compliance Model with Uncertainty Quantification

by George Balan, Elena Serea, Alexandru Sălceanu and Dorin-Dumitru Lucache

Mathematics 2026, 14(9), 1425; https://doi.org/10.3390/math14091425 - 23 Apr 2026

Cited by 1 | Viewed by 644

Abstract

Digital Twin (DT) implementations in electrical and industrial systems are governed by fragmented ISO/IEC and IEC standards spanning terminology, architecture, interoperability, lifecycle management, and cybersecurity. This paper proposes a mathematical framework that integrates these standards into a unified compliance model. A layered DT [...] Read more.

Digital Twin (DT) implementations in electrical and industrial systems are governed by fragmented ISO/IEC and IEC standards spanning terminology, architecture, interoperability, lifecycle management, and cybersecurity. This paper proposes a mathematical framework that integrates these standards into a unified compliance model. A layered DT architecture is defined as a finite set of functional abstractions, and standards are linked to layers through a multivalued mapping and an incidence matrix. Traceability, interoperability, fidelity, and security/governance indicators are normalized and aggregated through a bounded weighted functional to obtain a deterministic compliance score. The model is then extended by treating selected indicators as random variables, which enables probabilistic maturity classification and Monte Carlo-based robustness analysis. The resulting functional is bounded, monotone, and stable under bounded perturbations. Numerical experiments on a synthetic portfolio illustrate deterministic scoring and uncertainty effects. The framework provides a proof-of-concept basis for structured DT compliance assessment across heterogeneous electrical systems; however, broader empirical validation is still required before operational deployment. Full article

(This article belongs to the Special Issue Mathematical Applications in Electrical Engineering, 2nd Edition)

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18 pages, 855 KB

Open AccessArticle

Prediction of Remaining Life and Insulation Failure of High-Voltage Distribution Cable Using Statistical Methods

by Filip Zec, Saša D. Milić, Đorđe Lazarević and Jasna Dragosavac

Mathematics 2026, 14(7), 1164; https://doi.org/10.3390/math14071164 - 31 Mar 2026

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Abstract

Predicting the remaining life and insulation failure of high-voltage distribution cables using statistical methods is essential for ensuring the reliability and safety of electrical power systems. Statistical techniques enable the identification of degradation trends by analyzing historical operational and diagnostic data. The paper [...] Read more.

Predicting the remaining life and insulation failure of high-voltage distribution cables using statistical methods is essential for ensuring the reliability and safety of electrical power systems. Statistical techniques enable the identification of degradation trends by analyzing historical operational and diagnostic data. The paper examines the life expectancy of 10 kV Paper-Insulated Lead-Covered (PILC) cable insulation. The presented experiment was performed on both used and new cable samples. Presumptions of Weibull distribution parameters for random variables “breakdown voltage” and “breakdown time” are experimentally validated. The exponent of life expectancy for both used and new cable samples is obtained from experimentally derived parameters of the Weibull distribution. As a result, the dependence quantile for the breakdown probability of used and new cables is determined. Full article

(This article belongs to the Special Issue Mathematical Applications in Electrical Engineering, 2nd Edition)

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17 pages, 2265 KB

Open AccessArticle

Current Transformer Error Compensation Under Core Saturation Conditions Based on Machine Learning Algorithms

by Ismoil Odinaev, Svetlana Beryozkina, Andrey Pazderin, Mihail Senyuk, Murodbek Safaraliev and Pavel Dubrovin

Mathematics 2026, 14(3), 568; https://doi.org/10.3390/math14030568 - 5 Feb 2026

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Abstract

To provide information support for relay protection and emergency automation algorithms, electromagnetic measuring voltage and current transformers are most often used. As practice shows, the magnetic core of the current transformers can be saturated under transient processes. This negatively impacts the proper functioning [...] Read more.

To provide information support for relay protection and emergency automation algorithms, electromagnetic measuring voltage and current transformers are most often used. As practice shows, the magnetic core of the current transformers can be saturated under transient processes. This negatively impacts the proper functioning of protection systems. This paper proposes a methodology for restoration of the current transformers’ secondary current based on machine learning algorithms. The task of current restoration is reduced to clustering and regression problems. The groups’ current data are clustered depending on the depth of core saturation and the shape of current distortion. Then, solving the regression problem, current restoration is performed. Considering the requirements for the performance of the protection system, the following machine learning algorithms were selected for current recovery: Decision Tree, Random Forest, XGBoost, and Support Vector Machine for regression problems. The results of computational experiments show that the optimal number of clusters is four. Among the current restoration algorithms, XGBoost proved to be the most suitable. On average, for 17,240 test saturation modes, its error was 4%. The time delay for restoration one saturation mode was 0.0067 ms. Full article

(This article belongs to the Special Issue Mathematical Applications in Electrical Engineering, 2nd Edition)

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14 pages, 3496 KB

Open AccessArticle

Two-Dimensional Steady-State Thermal Analytical Model of Dual-PM Consequent-Pole Magnetically Geared Machine Based on Harmonic Modeling

by Manh-Dung Nguyen, Duy-Tinh Hoang, Kyung-Hun Shin, Kyong-Hwan Kim, Ji-Yong Park and Jang-Young Choi

Mathematics 2026, 14(3), 460; https://doi.org/10.3390/math14030460 - 28 Jan 2026

Viewed by 963

Abstract

This paper presents a mathematical approach for analyzing the thermal behavior of a dual-permanent-magnet consequent-pole magnetically geared machine. The analytical method, referred to as harmonic modeling, employs a complex Fourier series and the Cauchy product to obtain solutions to the partial differential equations [...] Read more.

This paper presents a mathematical approach for analyzing the thermal behavior of a dual-permanent-magnet consequent-pole magnetically geared machine. The analytical method, referred to as harmonic modeling, employs a complex Fourier series and the Cauchy product to obtain solutions to the partial differential equations governing the temperature distribution in electrical machines. Unlike lumped-parameter thermal networks that provide only average quantities, the proposed approach enables the prediction of spatial temperature distributions. The machine is further investigated under various operating conditions, including different convection coefficients and loss levels. An 11-pole, 18-slot prototype was evaluated by comparison with finite element method (FEM) simulations. The results demonstrate that the proposed method agreed well with the FEM results, with errors below 10%, while requiring less than 2 s per calculation compared with approximately 20 s for FEM simulations. Full article

(This article belongs to the Special Issue Mathematical Applications in Electrical Engineering, 2nd Edition)

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28 pages, 3571 KB

Open AccessArticle

Methodology for Transient Stability Assessment and Enhancement in Low-Inertia Power Systems Using Phasor Measurements: A Data-Driven Approach

by Mihail Senyuk, Svetlana Beryozkina, Ismoil Odinaev, Inga Zicmane and Murodbek Safaraliev

Mathematics 2025, 13(19), 3192; https://doi.org/10.3390/math13193192 - 5 Oct 2025

Viewed by 1356

Abstract

Modern energy systems are undergoing a profound transformation characterized by the active replacement of conventional fossil-fuel-based power plants with renewable energy sources. This transition aims to reduce the carbon emissions associated with electricity generation while enhancing the economic performance of electric power market [...] Read more.

Modern energy systems are undergoing a profound transformation characterized by the active replacement of conventional fossil-fuel-based power plants with renewable energy sources. This transition aims to reduce the carbon emissions associated with electricity generation while enhancing the economic performance of electric power market players. However, alongside these benefits come several challenges, including reduced overall inertia within energy systems, heightened stochastic variability in grid operation regimes, and stricter demands on the rapid response capabilities and adaptability of emergency controls. This paper presents a novel methodology for selecting effective control laws for low-inertia energy systems, ensuring their dynamic stability during post-emergency operational conditions. The proposed approach integrates advanced techniques, including feature selection via decision tree algorithms, classification using Random Forest models, and result visualization through the Mean Shift clustering method applied to a two-dimensional representation derived from the t-distributed Stochastic Neighbor Embedding technique. A modified version of the IEEE39 benchmark model served as the testbed for numerical experiments, achieving a classification accuracy of 98.3%, accompanied by a control law synthesis delay of just 0.047 milliseconds. In conclusion, this work summarizes the key findings and outlines potential enhancements to refine the presented methodology further. Full article

(This article belongs to the Special Issue Mathematical Applications in Electrical Engineering, 2nd Edition)

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19 pages, 9204 KB

Open AccessArticle

Numerical Study of Salt Ion Transport in Electromembrane Systems with Ion-Exchange Membranes Having Geometrically Structured Surfaces

by Evgenia Kirillova, Natalia Chubyr, Anna Kovalenko and Mahamet Urtenov

Mathematics 2025, 13(9), 1523; https://doi.org/10.3390/math13091523 - 6 May 2025

Cited by 1 | Viewed by 1203

Abstract

This article is devoted to numerically modeling the effect of the geometric modification of the surfaces of ion-exchange membranes in electromembrane systems (EMSs) on the salt ion transport using a 2D mathematical model of the transport process in the desalination channel based on [...] Read more.

This article is devoted to numerically modeling the effect of the geometric modification of the surfaces of ion-exchange membranes in electromembrane systems (EMSs) on the salt ion transport using a 2D mathematical model of the transport process in the desalination channel based on boundary value problems for the coupled system of Nernst–Planck–Poisson and Navier–Stokes equations. The main patterns of salt ion transport are established taking into account diffusion, electromigration, forced convection, electroconvection, and the geometric modification of the surface of ion-exchange membranes. It is shown that the geometric modification of the surface of ion-exchange membranes significantly changes both the formation and development of electroconvection. A significant combined effect of electroconvection and geometric modification of the surface of ion-exchange membranes in the desalination channel on the salt ion transport is shown, as well as a complex, nonlinear, and non-stationary interaction of all the main effects of concentration polarization in the desalination channel. Full article

(This article belongs to the Special Issue Mathematical Applications in Electrical Engineering, 2nd Edition)

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