Mathematical Modeling of Electrical Power Systems for Fault Detection

Introduction to Circuit Analysis and Design

Industrial companies are the largest electric power consumers in our country (about 60% of all generated power). Mining companies are key to the budget formation of the Republic of Sakha (Yakutia). They are, at the same time, very energy intensive. For instance, diamond, gold, and coal mining companies consume about 75% of all the generated power. Therefore, energy-saving technologies are getting more popular in designing the electric power systems for such enterprises. When designing electric power supply and drive systems for an industrial company, it is necessary to perform a lot of calculations and analyze a large number of diagrams. This process can be significantly simplified through the use of computer modeling software. During the design stage, it is safe to make mistakes in a computer model because the program can always identify the parameter or element that caused the problem. Currently, specific programs designed for the modeling of complex engineering systems can boast modeling accuracies of 90-95%. MatLab is one of such programs that can help develop both mathematical and physical models. Modeling uses real-life specifications of production plants and industrial companies. Modeling can provide information on such physical parameters of the electric power system as the current, voltage, and power levels, short-circuit currents, load schedules, quality of electricity, and deviations of modeled parameter values from the calculated ones, etc. Based on the calculations performed, we can provide recommendations for the power supply services of industrial companies and deploy energy-saving technologies.

Preface. 1. The Physics of Electricity. 1.1 Basic Quantities. 1.1.1 Introduction. 1.1.2 Charge. 1.1.3 Potential or Voltage. 1.1.4 Ground. 1.1.5 Conductivity. 1.1.6 Current. 1.2 Ohm's law. 1.2.1 Resistance. 1.2.2 Conductance. 1.2.3 Insulation. 1.3 Circuit Fundamentals. 1.3.1 Static Charge. 1.3.2 Electric Circuits. 1.3.3 Voltage Drop. 1.3.4 Electric Shock. 1.4 Resistive Heating. 1.4.1 Calculating Resistive Heating. 1.4.2 Transmission Voltage and Resistive Losses. 1.5 Electric and Magnetic Fields. 1.5.1 The Field as a Concept. 1.5.2 Electric Fields. 1.5.3 Magnetic Fields. 1.5.4 Electromagnetic Induction. 1.5.5 Electromagnetic Fields and Health Effects. 1.5.6 Electromagnetic Radiation. 2. Basic Circuit Analysis. 2.1 Modeling Circuits. 2.2 Series and Parallel Circuits. 2.2.1 Resistance in Series. 2.2.2 Resistance in Parallel. 2.2.3 Network Reduction. 2.2.4 Practical Aspects. 2.3 Kirchhoff's Laws. 2.3.1 Kirchhoff's Voltage Law. 2.3.2 Kirchhoff's Current Law. 2.3.3 Application to Simple Circuits. 2.3.4 The Superposition Principle. 2.4 Magnetic Circuits. 3. AC Power. 3.1 Alternating Current and Voltage. 3.1.1 Historical Notes. 3.1.2 Mathematical Description. 3.1.3 The rms Value. 3.2 Reactance. 3.2.1 Inductance. 3.2.2 Capacitance. 3.2.3 Impedance. 3.2.4 Admittance. 3.3 Power. 3.3.1 Definition of Electric Power. 3.3.2 Complex Power. 3.3.3 The Significance of Reactive Power. 3.4 Phasor Notation. 3.4.1 Phasors as Graphics. 3.4.2 Phasors as Exponentials. 3.4.3 Operations with Phasors. 4. Generators. 4.1 The Simple Generator. 4.2 The Synchronous Generator. 4.2.1 Basic Components and Functioning. 4.2.2 Other Design Aspects. 4.3 Operational Control of Synchronous Generators. 4.3.1 Single Generator: Real Power. 4.3.2 Single Generator: Reactive Power. 4.3.3 Multiple Generators: Real Power. 4.3.4 Multiple Generators: Reactive Power. 4.4 Operating Limits. 4.5 The Induction Generator. 4.5.1 General Characteristics. 4.5.2 Electromagnetic Characteristics. 4.6 Inverters. 5. Loads. 5.1 Resistive Loads. 5.2 Motors. 5.3 Electronic Devices. 5.4 Load from the System Perspective. 5.4.1 Coincident and Noncoincident Demand. 5.4.2 Load Profiles and Load Duration Curve. 5.5 Single- and Multiphase Connections. 6. Transmission and Distribution. 6.1 System Structure. 6.1.1 Historical Notes. 6.1.2 Structural Features. 6.1.3 Sample Diagram. 6.1.4 Topology. 6.1.5 Loop Flow. 6.1.6 Stations and Substations. 6.1.7 Reconfiguring the System. 6.2 Three-Phase Transmission. 6.2.1 Rationale for Three Phases. 6.2.2 Balancing Loads. 6.2.3 Delta and Wye Connections. 6.2.4 Per-Phase Analysis. 6.2.5 Three-Phase Power. 6.2.6 D.C. Transmission. 6.3 Transformers. 6.3.1 General Properties. 6.3.2 Transformer Heating. 6.3.3 Delta and Wye Transformers. 6.4 Characteristics of Power Lines. 6.4.1 Conductors. 6.4.2 Towers, Insulators, and Other Components. 6.5 Loading. 6.5.1 Thermal Limits. 6.5.2 Stability Limit. 6.6 Voltage Control. 6.7 Protection. 6.7.1 Basics of Protection and Protective Devices. 6.7.2 Protection Coordination. 7. Power Flow Analysis. 7.1 Introduction. 7.2 The Power Flow Problem. 7.2.1 Network Representation. 7.2.2 Choice of Variables. 7.2.3 Types of Buses. 7.2.4 Variables for Balancing Real Power. 7.2.5 Variables for Balancing Reactive Power. 7.2.6 The Slack Bus. 7.2.7 Summary of Variables. 7.3 Example with Interpretation of Results. 7.3.1 Six-Bus Example. 7.3.2 Tweaking the Case. 7.3.3 Conceptualizing Power Flow. 7.4 Power Flow Equations and Solution Methods. 7.4.1 Derivation of Power Flow Equations. 7.4.2 Solution Methods. 7.4.3 Decoupled Power Flow. 7.5 Applications and Optimal Power Flow. 8. System Performance. 8.1 Reliability. 8.1.1 Measures of Reliability. 8.1.2 Valuation of Reliability. 8.2 Security. 8.3 Stability. 8.3.1 The Concept of Stability. 8.3.2 Steady-State Stability. 8.3.3 Dynamic Stability. 8.3.4 Voltage Stability. 8.4 Power Quality. 8.4.1 Voltage. 8.4.2 Frequency. 8.4.3 Waveform. 9. System Operation, Management, and New Technology. 9.1 Operation and Control on Different Time Scales. 9.1.1 The Scale of a Cycle. 9.1.2 The Scale of Real-Time Operation. 9.1.3 The Scale of Scheduling. 9.1.4 The Planning Scale. 9.2 New Technology. 9.2.1 Storage. 9.2.2 Distributed Generation. 9.2.3 Automation. 9.2.4 FACTS. 9.3 Human Factors. 9.3.1 Operators and Engineers. 9.3.2 Cognitive Representations of Power Systems. 9.3.3 Operational Criteria. 9.3.4 Implications for Technological Innovation. 9.4 Implications for Restructuring. Appendix: Symbols, Units, Abbreviations, and Acronyms. Index.

The topological (graph) structure of complex networks often provides valuable information about the performance and vulnerability of the network. However, there are multiple ways to represent a given network as a graph. Electric power transmission and distribution networks have a topological structure that is straightforward to represent and analyze as a graph. However, simple graph models neglect the comprehensive connections between components that result from Ohm's and Kirchhoff's laws. This paper describes the structure of the three North American electric power interconnections, from the perspective of both topological and electrical connectivity. We compare the simple topology of these networks with that of random (Erdos and Renyi, 1959), preferential-attachment (Barabasi and Albert, 1999) and small-world (Watts and Strogatz, 1998) networks of equivalent sizes and find that power grids differ substantially from these abstract models in degree distribution, clustering, diameter and assortativity, and thus conclude that these topological forms may be misleading as models of power systems. To study the electrical connectivity of power systems, we propose a new method for representing electrical structure using electrical distances rather than geographic connections. Comparisons of these two representations of the North American power networks reveal notable differences between the electrical and topological structure of electric power networks.

This paper proposes a comprehensive simulation model for the Electrical Power System (EPS) of More Electrical Aircrafts (MEA). Taking the presently typical MEA model of Boeing 787 as an example, the integrated structure of EPS is firstly illustrated and the operating principles are theorized. The key components of the MEA EPS including generators, rectifiers, loads, breakers, control devices, cables and other supplementary elements are then simulated to build a comprehensive EPS model on the platform of MATLAB/Simulink. Although the model's scale is large, its simulation time is guaranteed thanks to using the built-in blocks provided by Simulink. Finally, the stability analysis in terms of normal operation and different faults are carried out in case studies and power source switching strategies in different situations are summarized. The results showed that the comprehensive model proposed by the paper can be used to simulate MEA EPS and analyze its operational stability effectively and efficiently.

The use of series compensation devices is one of the methods to increase the capacity of a power transmission line. In addition, the use of controlled series compensation devices affects the steady-state stability of the electric power system. It is also widely known that automatic excitation control of generators has a positive effect on the stability of the system. Thus, it is reasonable to analyze the influence of the adjustable parameters of the controlled series compensation device on the line capacity when considering modes at the boundaries of the stability regions, taking into account automatic excitation controllers on synchronous generators of the power plant. Methods of mathematical modeling of the electric power system, the theory of long-distance power lines and electromechanical transients, methods of analyzing the stability of electric power systems are used. The original software in the C ++ programming language is used as a modeling tool. The authors have selected the regulation laws parameters of controlled series compensation device and automatic excitation controller, provided that steady-state stability is maintained. The stability regions of the electric power-engineering system under study are constructed depending on the tuning parameters of the considered devices. A technique is proposed to select the coefficients of complex regulation of the devices to increase the capacity of a high-voltage power transmission line. The proposed method of the complex selection of regulation coefficients of controlled devices provides the possibility of obtaining their optimal values in a controlled electric power system to ensure stability.

The article is devoted to the development and testing of a methodology for analytical modeling of power supply systems that include multiple sources and multiple consumers of electricity, located at a distance from each other and connected by power transmission lines. The developed methodology will help designers assess the possibilities of distributing electricity among consumers while calculating the optimal voltages that need to be maintained at the nodes of the power sources, thereby achieving minimal voltage differences between the load nodes. The methodology is tested using a 10-node power supply system, in which the number of energy source nodes and load nodes is equal to 5. The modeling methodology is based on calculating the parameters of the power supply system using the node potential method, along with Ohm's and Kirchhoff's laws. The calculations utilize the method of complex amplitudes and are carried out according to a proposed iterative algorithm, the flowchart of which is presented in the article. The calculations are performed in the MathCAD environment using the mathematical framework of matrix computations. For this purpose, a conductance matrix and a node current matrix are constructed, and the node potential matrix is calculated in MathCAD. The paper demonstrates how to transit from a three-phase system to a single-phase system, with all calculations performed on a single-phase scheme. To obtain accurate results, formulas are derived for converting the parameters of the three-phase system into single-phase parameters, including the conversion of consumed load powers into complex impedances. The results of the modeling are presented in tabular form and as directed graphs, indicating the potentials of the nodes and the directions of active power flows. The results of modeling two variants of power supply systems are provided. The proposed methodology is recommended for assessing the parameters of power supply systems that include multiple sources of electricity and several load nodes. Keywords: power supply system, mathematical modeling, calculation methodology, node potentials, complex amplitudes.

This paper presents the mathematical modeling and analysis of High voltage high pulse power system (HVHPPS) performance on various type of load like magnetron, klystron, particle accelerator etc. The performance of high voltage high pulse power supply changes with the change in the load parameters i.e. dynamic impedance, cable length, temperature variation, cavity dimension change, applied cathode voltage, magnetic strength etc. The high voltage high pulse power supply for RF and Microwave types of load requires always a very good impedance matching, to achieve maximum power transmission and minimum reflection. In this paper the mathematical modeling of subsystem of High Voltage High Pulse Power System and load are generated. Accordingly the load behavior changes incorporated and analyzed the effect at the source side based on hardware results obtained years together. The mathematical modeling analysis result is validated through the experimental hardware model.

Enhancing transmission line capacity is a critical challenge in modern power engineering. One of the promising approaches in this field involves the implementation of series compensation devices, which significantly impact power system static stability. The small-signal stability analysis method is widely employed for assessing power system stability. This technique formulates system motion differential equations under minor disturbances of the initial state and examining the resulting free oscillation patterns. A crucial step of this method is deriving the characteristic equation, which roots determine system stability. However, for complex multi-machine systems, particularly with automatic control of excitation of generators and series compensation devices, the characteristic equation becomes high-order, complicating direct analysis. Consequently, studying the characteristic equation's constant term gains particular importance, as its sign is critically significant to determine static aperiodic stability. Methods of mathematical modeling of the electric power system, the theory of long-distance power lines and electromechanical transients, automatic control theory for power systems, as well as methods of analyzing the stability of electric power systems have been used. The original software in C ++ programming language has been used as a modeling tool. A system of differential equations describing electromagnetic transients in the studied power system has been developed. Based on the initial nonlinear model, a first-approximation linearized system of equations has been obtained using the small-signal method, accounting for the mutual influence of generators as well as parameters of automatic excitation regulators and series compensation devices. To analyze the stability of the system, a characteristic determinant has been formulated and a corresponding characteristic equation of the eighth order has been obtained. The dependence of the constant term values on the angle  between the EMFs of the two generators has been established. It has been proven analytically using numerical methods that the final expression for a constant term does not directly contain the tuning parameters of the automatic excitation controllers. The obtained result has significant practical value. Since, on the one hand, it reduces the required symbolic computations and simplifies mathematical models and on the other hand, it completely maintains accuracy and validity in the system's stability assessment. The co-use of automatic excitation controllers and series compensation devices also requires coordinated tuning.

In this paper is presented new method of an integral transformation of alternating periodic functions for the determination of circuit parameters of an electromagnetic phenomenon. This integral transformation is based on the electromagnetic physical laws: Ohm's law, induction law, equation of continuity, Kirchhoff's laws and the mathematical definition of the scalar (inner) product time varying quantities. The new method is an effective tool for the production of the circuit models of electromagnetic phenomena.

These proceedings collect the academic and scientific contributions presented at the X International Conference Days of Mathematical Modeling in Sciences (X ICDMMS), which continues the legacy of the “Days of Applied Mathematics” (ICDAM) previously held at Universidad Simón Bolívar, San José de Cúcuta, Colombia. The X ICDMMS took place in Talca, Chile, on October 2–25, 2024. The primary objective of this conference is to promote the exchange of concrete experiences in applying mathematical knowledge to address complex problems across diverse scientific contexts. In line with this mission, the event has brought together a vibrant academic and scientific community, including postgraduate students, professors, and researchers from across Latin America, who have not only presented their research but have also had the opportunity to publish their work (since the fourth edition) in the Journal of Physics: Conference Series, the journal associated with the conference. This event is an initiative led by three academic research groups with a presence in Chile and Colombia: the Mathematical Modeling Group of Maule (GM3), the Scientific Modeling and Business Innovation Group of Universidad Simón Bolívar (GIMCINE), and the Mathematical Modeling Group of Biological Systems at Universidad Tecnológica Metropolitana (MATBIO-UTEM). Its purpose is to serve as a platform for disseminating knowledge in the basic and applied sciences through the lens of mathematical modeling. A key focus of the conference is to continue encouraging a space for the scientific contributions of graduate students and to promote interactions that strengthen their integration into the research community. The topics developed in the X ICDMMS were the following: (i) Computer Science in Mathematics and Engineering (ii) Modeling in Basic Sciences (iii) Modeling in Social Sciences (iv) Mathematical Biology (v) Modeling, Simulation, and Diagnostics (vi) Education and Communication in Science and Engineering The ICDMMS forms an integral part of the training and research processes within the academic programs in Applied Mathematical Modeling, Mathematics Education, and Experimental Sciences Education at the Faculty of Basic Sciences of the Universidad Católica del Maule. It aligns with the institution's commitment to high-quality education and responds to the demands of the digital era by emphasizing the modeling of scientific knowledge across diverse fields, including mathematics, physics, biology, and chemistry, as well as applications in medicine, engineering, management, business, and education. The ICDMMS offers emerging researchers opportunities for participation and growth through meaningful interactions and dialogue with established researchers from various scientific disciplines represented at the conference. It also allows students, faculty, and researchers a valuable space for professional development and knowledge acquisition through thematic sessions. List of Organizing Committee, National Scientific Committee, International Scientific Committee, Organizing groups and partners and Sponsors are available in this PDF.

One of the trends in the global energy sector is the penetration of distributed generating units (distributed generation - DG). The ongoing transformation of electric power systems (EPS) due to this leads to a significant change in the properties of power systems. Problems arise with ensuring the main condition for the reliability and survivability of energy regions with DG and EPS units as a whole - sustainability. Moreover, one of the main tasks remains to provide it with great indignation. The main direction of solving this problem is mathematical modeling. However, the necessary mathematical model EPS of real dimension contains a rigid nonlinear system of differential equations of an extremely high order. To improve the conditionality of the EPS mathematical model in numerical integration, simplifications and restrictions are inevitably used, which reduce the completeness and reliability of the obtained simulation results. In this regard, it becomes necessary to validate them. The article proposes an alternative to the existing direction of validation, based on the use of a model standard instead of field data. This validation makes it possible to reveal the influence of the applied simplifications and restrictions in the numerical simulation on the quality of solving the problems of estimating the stability of EPS with DG under a large disturbance. The studies presented in the article were carried out using the example of an energy region with DG units, which is part of an EPS of real dimension, and demonstrate the arising errors in calculating stability with a large disturbance, the nature of their change and the reasons for their occurrence, as well as influencing factors.

The analysis of existing mathematical models for power systems with large shares of electricity production by NPP and renewable energy sources generation was carried out. The low level of adequacy is shown for power system models used in the problems of long-term development planning for generating plants with nuclear fuel and renewable energy sources. In order to plan the development of power systems at the current stage of technological changes in the electric power industry, new models of load modes for NPP units and generating equipment with renewable energy sources are proposed. Dispatching models with a cyclic forecast period were used to ensure the adequacy unit commitment models for NPP units and energy storage systems. Representative models of solar and wind power plants are presented, which, together with cluster models of NPP units and energy storage systems, solve the issue on the adequacy of power system modeling in current conditions of their development. The results of computational experiments with the proposed models for generating equipment of the flexible grid are presented. The adequacy of cluster mathematical models of NPP units and energy storage systems, as well as representative models of solar and wind power plants generating equipment is experimentally confirmed.

This paper presents a mathematical modeling approach by which to solve the power flow and state estimation problems in electric power systems through a mathematical programming language (AMPL). The main purpose of this work is to show the advantages of representing these problems through mathematical optimization models in AMPL, which is a modeling language extensively used in a wide range of research applications. The proposed mathematical optimization models allow for dealing with particular issues in that they are not usually considered in the classical approach for power flow and state estimation, such as solving the power flow problem considering reactive power limits in generation buses, as well as the treatment of errors in state estimation analysis. Furthermore, the linearized mathematical optimization models for both problems at hand are also presented and discussed. Several tests were carried out to validate the proposed optimization models, evidencing the applicability of the proposed approach.

Software time-domain simulation models are useful to the naval engineering community both for the system design of future vessels and for the in-service support of existing vessels. For future platforms, the existence of a model of the vessel's electrical power system provides a means of assessing the performance of the system against defined requirements. This could be at the stage of requirements definition, bid assessment or any subsequent stage in the design process. For in-service support of existing platforms, the existence of a model of the vessel's electrical power system provides a means of assessing the possible cause and effect of operational defects reported by ship's staff, or of assessing the possible future implications of some change in the equipment line-up or operating conditions for the vessel. Detailed high fidelity time-domain simulation of systems, however, can be problematic due to extended execution time. This arises from the model's mathematically stiff nature: models of Integrated Electric Propulsion systems can also require significant computational resource. A conventional time-domain software simulation model is only able to utilize a single computer processor at any one time. The duration of time required to obtain results from a software model could be significantly reduced if more computer processors were utilized simultaneously. This paper details the development of a distributed simulation environment. This environment provides a mechanism for partitioning a time-domain software simulation model and running it on a cluster of computer processors. The number of processors utilized in the cluster ranges between four and sixteen nodes. The benefit of this approach is that reductions in simulation duration are achievable by an appropriate choice of model partitioning. From an engineering perspective, any net timing reduction translates to an increase in the availability of data, from which more efficient analysis and design follows.