Control of Overhead Power Lines with Unmanned Aerial Vehicles (UAVs) (Studies in Systems, Decision and Control, 359) 3030697517, 9783030697518

Table of contents :
Introduction
Contents
Abbreviations
Monitoring of Energy Objects Parameters with Using UAVs
References
Improving Methods for One-Sided Determination of the Location of Damage of Overhead Power Lines in Networks with Effectively Grounded Neutral Based on UAVs
1 Mathematical Model of One-Sided Determination of the LoD of OPL by EMP
1.1 Single-Phase SC
1.2 Two-Phase SC
1.3 Three-Phase SC
2 Methodology for Determining the LoD of OPL by Mode Parameters Obtained on One of Its Sides
2.1 Calculation of System Parameters of the Opposite End of the Line
2.2 Calculation of SC Currents from the System of the Opposite End of the Line
2.3 Method for Determining the Distance to the Damage Place
References
Automation of Determining the Location of Damage of Overhead Power Lines
1 Functional Purpose of the BAC
2 Selection of the Start Time and Duration of the Emergency Analysis Interval
2.1 Determination of the SC Beginning
2.2 Minimize Emergency Analysis Interval
3 Determination of the Type of SC and Special Phase
4 Determination of Location of SC
References
Experimental Studies of the Method for Determining Location of Damage of Overhead Power Lines in the Operation Mode
1 Hardware-Software Complex “TsPRS”
2 Analysis of Emergency Blackouts of Power Lines in 110 kV Networks
2.1 Damage to the “L-SiM” Overhead Line
2.2 Damage of the “I-S” Overhead Line
2.3 Damage of the “K-B” Overhead Line
References
Mathematical Models of Electric Fields of Electric Transmission Lines
1 Analytical Methods for EFs Modeling of OPLs
References
Physical Modeling of Discharges in Long Air Gaps with the Presence of the Corona at the Tops of Grounded Objects
1 Description of the High-Voltage Experimental Bench and Research Methodology
2 Results of Experimental Studies of Breakdown Processes in the Presence of Corona Discharge at the Tops of Grounded Rods
References
Mathematical Modeling of the Electromagnetic Processes of the Corona’s Formation During the Operation of Electric Power Facilities
1 Modeling of EFs in the Presence of Rods with Rounded Tops
1.1 Modeling of EFs in the Presence of Curved Interface Surfaces
1.2 Calculation of EF in the Vicinity of an Electrically Conductive Cylindrical Rod
1.3 Influence of the Geometry of Rods with Rounded Tops at the Level of Maximum EF Strength
2 Combined Method of Mathematical Modeling of EF Amplification on Rounded Vertices of Very Long Cylindrical Rods
2.1 Calculation of EF of a Cylindrical Long Rod with a Step of the Computational Grid Proportional to Its Radius
2.2 Calculation of EF of a Long Cylindrical Rod with a Step of the Computational Grid Proportional to Its Length
2.3 Calculation of EF Amplification at the Tops of Long Cylindrical Rods
References
Physical Modeling of the Electrophysical Processes of the Formation of the Corona During the Operation of Electric Power Facilities
1 Physical Modeling of Electromagnetic Processes During the Development of the Corona on Rod Electrodes with Vertices of Various Shapes
2 Mathematical Modeling of Corona Formation Processes at the Tops of Rod Electrodes
References
Acoustic Diagnostics for Determining the Appearance of Corona Discharge
1 Relationship of the Qualitative Parameters of Electrical Energy and Corona Discharge
2 Acoustic Noises in Electrical Installations and Corona Discharge
3 Experimental Studies of the Acoustic Component of the Corona Discharge
3.1 Acoustic Measurements and Their Results
3.2 Spectral Decomposition of Audio Corona Noise Fragments
4 Recognition of a Corona Discharge in the Presence of Spectral Components
5 Acoustic Corona Discharge Field Construction
5.1 Construction of a Curve of Acoustic Strength from Corona Discharge Elements
5.2 Calculation of the Acoustic Field from Elements with a Corona Discharge
5.3 Calculation of the Acoustic Field from a Corona Wire
References

Citation preview

Studies in Systems, Decision and Control 359

Yevgen I. Sokol Artur O. Zaporozhets   Editors

Control of Overhead Power Lines with Unmanned Aerial Vehicles (UAVs)

Studies in Systems, Decision and Control Volume 359

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

More information about this series at http://www.springer.com/series/13304

Yevgen I. Sokol · Artur O. Zaporozhets Editors

Control of Overhead Power Lines with Unmanned Aerial Vehicles (UAVs)

Editors Yevgen I. Sokol National Technical University “Kharkiv Polytechnic Institute” Kharkiv, Ukraine

Artur O. Zaporozhets Department of Monitoring and Optimization of Thermophysical Processes Institute of Engineering Thermophysics of NAS of Ukraine Kyiv, Ukraine

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-69751-8 ISBN 978-3-030-69752-5 (eBook) https://doi.org/10.1007/978-3-030-69752-5 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Introduction

Preserving the safety and energy efficiency of the functioning of the United Energy System (UES) of Ukraine with minimal energy consumption is one of the most important conditions for ensuring the country is energy independent. Any failures and malfunctions during the operation of electric power transmission systems from a producer to a consumer lead to significant losses of electric power. Hence, the need for the speediest restoration of the performance of the UES elements after accidents exists. At the same time, monitoring the state of power transmission systems from producer to consumer is of particular importance. Such monitoring is aimed at preventing blackouts of power transmission lines and accidents at high-voltage substations, hydro, heat, and nuclear power plants, as well as the quick restoration of their performance. The most promising modern method for diagnosing the state of energy facilities, which requires significantly lower energy consumption compared to existing ones, and also allows minimizing the recovery time of the UES elements after accidents, is monitoring using Unmanned Aerial Vehicles (UAVs). At the same time, the condition of the research objects is controlled by optical registration, registration with the help of thermal imagers, means of measuring acoustic signals, as well as registration of partial discharges that occur in isolation, which is performed automatically without involving expensive maintenance equipment. The energy efficiency of the UES of Ukraine substantially depends on the operational transmission of information on the distribution of stress values along the power transmission lines and the nature of weather conditions. The use of UAVs for monitoring and fixing these parameters along the length of power transmission lines allows to reduce losses during energy transmission by regulating the voltage of individual sections of electric networks. In addition, the availability of such data obtained in real time can significantly improve the accuracy of short-term forecasting of their operation mode and improve the dispatch control of the energy system as a whole during calculating network modes. Thus, the development of a set of methods to improve the safety of functioning and energy efficiency of the UES of Ukraine using UAVs is an urgent scientific and technical task. v

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Operational monitoring of the status of power transmission lines, as well as highvoltage substations, hydroelectric power stations, thermal power plants, and nuclear power plants, is an integral part of the system to ensure the reliability of energy supply and eliminate the causes of emergency outages. UAV motion control during monitoring of the safety of energy system facilities can be done by measuring the strength of the Electric (EF) or Magnetic (MF) field. Analysis of these parameters using the developed mathematical models is also used to optimize the operating modes and structure of the research objects. The most common damage of power transmission lines is wire breakage occurring due to icing, crossing, breakdowns of insulators, etc., which leads to Short Circuits (SC) on power transmission lines. For quick elimination of an accident, it is necessary to accurately determine the place of damage. The approximate location and type of SC are determined using the developed hardware and algorithms. Then this information is entered into the UAV control system, with the help of which the exact location of the damage is determined with the optimization of the route of the operational on-site repair team. Given that access to power transmission lines to determine the location of damage is difficult in some cases and is a laborious and lengthy process, the algorithm for finding the location of damage using UAVs is very effective. In practice, it is very difficult or even impossible to measure the parameters of electromagnetic fields inside various technical devices, including UAVs, especially in emergency conditions. To solve this problem, methods of mathematical modeling of three-dimensional electromagnetic fields in inhomogeneous media were developed and software implemented. Such models, in particular, make it possible to obtain reference values of the distributions of EF and MF, and can be compared with measured UAVs. Yevgen I. Sokol Artur O. Zaporozhets

Contents

Monitoring of Energy Objects Parameters with Using UAVs . . . . . . . . . . . Marina M. Rezinkina, Yevgen I. Sokol, Artur O. Zaporozhets, Oleg G. Gryb, Ihor T. Karpaliuk, and Sergiy V. Shvets Improving Methods for One-Sided Determination of the Location of Damage of Overhead Power Lines in Networks with Effectively Grounded Neutral Based on UAVs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gennadiy A. Senderovich, Artur O. Zaporozhets, Oleg G. Gryb, Ihor T. Karpaliuk, Sergiy V. Shvets, and Natalia V. Rudevich Automation of Determining the Location of Damage of Overhead Power Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gennadiy A. Senderovich, Artur O. Zaporozhets, Oleg G. Gryb, Ihor T. Karpaliuk, Sergiy V. Shvets, and Inna A. Samoilenko Experimental Studies of the Method for Determining Location of Damage of Overhead Power Lines in the Operation Mode . . . . . . . . . . Gennadiy A. Senderovich, Artur O. Zaporozhets, Oleg G. Gryb, Ihor T. Karpaliuk, Sergiy V. Shvets, and Inna A. Samoilenko Mathematical Models of Electric Fields of Electric Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marina M. Rezinkina, Yevgen I. Sokol, Artur O. Zaporozhets, Oleg G. Gryb, Ihor T. Karpaliuk, and Sergiy V. Shvets Physical Modeling of Discharges in Long Air Gaps with the Presence of the Corona at the Tops of Grounded Objects . . . . . . Marina M. Rezinkina, Yevgen I. Sokol, Artur O. Zaporozhets, Oleg G. Gryb, Ihor T. Karpaliuk, and Sergiy V. Shvets Mathematical Modeling of the Electromagnetic Processes of the Corona’s Formation During the Operation of Electric Power Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marina M. Rezinkina, Yevgen I. Sokol, Artur O. Zaporozhets, Oleg G. Gryb, Ihor T. Karpaliuk, and Sergiy V. Shvets

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Physical Modeling of the Electrophysical Processes of the Formation of the Corona During the Operation of Electric Power Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Marina M. Rezinkina, Yevgen I. Sokol, Artur O. Zaporozhets, Oleg G. Gryb, Ihor T. Karpaliuk, and Sergiy V. Shvets Acoustic Diagnostics for Determining the Appearance of Corona Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Oleg G. Gryb, Ihor T. Karpaliuk, Artur O. Zaporozhets, Sergiy V. Shvets, and Natalia V. Rudevich

Abbreviations

ACR BAC BDPCS CAP CDEMP DLB DLoD DR EF EMF EMP GIS LoD MF OPL PML PVG SC sub UAV UES

Automatic circuit recloser Block of automation of calculations Block of determining of parameters of C2 system Curve of acoustic power Calculation device of emergency mode parameters Damage location block Device for location of damage Digital recorder Electric field Electromotive force Emergency mode parameters Geographic information system Location of damage Magnetic field Overhead power line Perfectly matched layers Pulsed voltage generator Short circuit Substitution Unmanned aerial vehicle United energy system

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Monitoring of Energy Objects Parameters with Using UAVs Marina M. Rezinkina , Yevgen I. Sokol , Artur O. Zaporozhets , Oleg G. Gryb , Ihor T. Karpaliuk , and Sergiy V. Shvets

Abstract The chapter describes the causes of overhead power line failures. It was noted that a significant number of power line emergency shutdowns occurs due to vandalism, such as the destruction of insulator strings, the theft of wires and support elements of power transmission lines, drafts on power transmission lines, etc. Thus, it is shown that monitoring the state of electric power facilities can significantly reduce the number of failures and prevent reasons for their occurrence. The concept of the state of electric power facilities is specified. It is proposed to use the registration of radiation: electric fields, electromagnetic fields, optical radiation. Registration of radiation can be performed at a distance from the object, which allows to use UAVs for monitoring. The authors structured and described the tasks in the energy sector, which can be solved with the help of UAVs. The classification of UAVs is shown. The classification of tasks in the energy sector associated with UAV types as the most suitable from the point of view of the tasks being solved is given. Keywords Overhead power line · Electric power facilities · Unmanned aerial vehicle · UAV · UAVs classification · Emergency shutdowns · Monitoring tasks

M. M. Rezinkina Department of Theoretical Electrical Engineering, National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine Y. I. Sokol National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine e-mail: [email protected] A. O. Zaporozhets (B) Department of Monitoring and Optimization of Thermophysical Processes, Institute of Engineering Thermophysics of NAS of Ukraine, Kyiv, Ukraine e-mail: [email protected] O. G. Gryb · I. T. Karpaliuk · S. V. Shvets Department of Automation and Cybersecurity, National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 Y. I. Sokol and A. O. Zaporozhets (eds.), Control of Overhead Power Lines with Unmanned Aerial Vehicles (UAVs), Studies in Systems, Decision and Control 359, https://doi.org/10.1007/978-3-030-69752-5_1

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In this chapter, it will be considering methods of integrated automated monitoring of objects of the energy system of Ukraine aimed at ensuring the safe functioning of its equipment and personnel. Some of them, that are currently used for monitoring and diagnostics the state of power equipment, are described in these works [1–8]. As the analysis of the causes of OPL failures shows, they mainly occur on 110 kV OPLs—86%, 11%—on 220 kV OPLs and 3% on 330–750 kV OPLs [9]. This distribution of the number of failures is proportional to the length of the corresponding OPLs. The largest number of OPLs’ emergency shutdowns is caused by damage of wires and lightning protection cables—56%. Other reasons for disconnecting OPLs are such damage as breakdowns of insulators—19%, damage of poles—15% and other elements of OPLs—10%. According to available statistics, a sharp increase in OPL outages occurs in the spring-summer period due to overlapping isolation gaps by green spaces. Also, a significant number of OPL emergency shutdowns occurs due to vandalism, such as the destruction of insulator strings, the theft of wires and power pylons elements of OPLs, hanging external objects on power transmission lines, etc. [10]. Monitoring the state of OPLs means a set of remote methods for studying the facility, which are carried out at a distance, without direct contact with it. Information is usually transmitted using electromagnetic radiation and is characterized by parameters such as spectral composition, intensity and direction of propagation. By registering these parameters, which also depend on the physical properties and condition of the object, as well as its spatial position, it can be studying it remotely. In this case, the simultaneous registration of radiation in several spectral zones allow to get the most versatile characteristics of the object. Radiation can be recorded both at individual points on the surface and along a path or area depending on the equipment used [11]. Registration of radiation is carried out using special equipment: photographic, optoelectronic, electronic. In most cases, it is not individual images that are recorded and analyzed, but their series or continuous sequences (video stream), which may differ in angle, visibility, shooting time and spectral range, which ensures a more detailed and accurate study of the object. It is also possible to measure the strength of the EP and MP fields along the span path. Aging OPLs leads to a decrease in its throughput and, as a result, to a limitation of electricity supplies to the end consumer. One of the main reasons for this is the degradation of the linear dimensions of the OPLs [12], which is due to: • • • •

change in the relief of the underlying surface; construction of various structures; overgrowing and increasing the cultural layer; construction and reconstruction of railway and transport routes.

If we turn to international experience in solving these problems, it can be noted that in most developed countries with an extensive electric power infrastructure (USA, EU countries [13], China [14], Brazil [15], Australia and New Zealand), activities are being taken to monitor and evaluate the network capacity, as well as activities for increasing their throughput.

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Network monitoring and inspection activities consist of the following steps: 1. 2. 3. 4. 5. 6.

7. 8.

Creation of three-dimensional models of OPLs and infrastructure elements with topographic reference. Thermal and visual inspection of lines to detect increased heating of contact compounds. Analysis of overall distances to the underlying surface, objects and vegetation. Identification of spans and structural elements that limit throughput or have violated overall distances. Inspection of power pylons and places of their fastening in the ground (foundations, metal structures, grounding devices). Instrumental control of the residual cross-section of wires and ground wire through selective scanning using non-destructive testing technologies (magnetic flaw detectors). Operational control of OPLs in case of emergency. Determining the facts of unauthorized access.

Based on the results of monitoring and analysis of networks, the following parameters are determined: • line capacity: permissible currents and temperature of wires; • problematic sections of lines, power pylons and foundations; • the magnitude of the tension of wires and ground wire. After the examination, activities are taken to increase the line’s throughput. World experience shows that such projects have high investment and practical value, and the reliability and safety of the OPL operation significantly increase when they are implemented [16–19]. Thus, one of the main tasks of monitoring OPLs are: determining the dimension of sagging wires, determining places of increased heating and overgrowing of OPLs with vegetation. Scientific and technological achievements of recent years allow to use both classical methods and fundamentally new approaches to solving these problems, for example, topographic monitoring using UAVs, which is based on laser, visual and thermal imaging scanning (location) of OPLs with subsequent photogrammetric processing of the data. This allows to determine the position, size, shape and condition of the monitoring object, as well as the nature of the change in these parameters over time [20–22]. The use of modern methods and algorithms of location allows to quickly obtain accurate maps of the OPLs’ routes, as well as the position of all its parts, to determine the topology of the terrain and the location of other objects that are in close proximity to the route. Modern methods provide a three-dimensional (3D) model of the route, as well as any geometric measurements using this model [23–25]. The use of specially developed software allows to compare multispectral images recorded with the help of UAVs with pre-created samples corresponding to the standard operating modes of the controlled OPLs. Such reference images can be combined into a single document—an OPL’s passport. It should also contain measured and calculated values of the voltage levels of the EF and MF at the places

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where the personnel of energy facilities are located and the equipment most vulnerable to electromagnetic interference. If necessary, within the framework of the monitoring, recommendations are provided about the design and location of electromagnetic shields that reduce the levels of electromagnetic influences and lightning rods, reduce the likelihood of lightning striking of the studied electric power objects. This book presents analytical expressions that formed the basis of the developed software for calculating the EF’s strength in the vicinity of OPLs. This software can also be used to navigate UAVs along OPLs and to recognize violations of its functioning. A comparison of the dependences of the EF’s strength calculated using this software with publicly available data shows that the key parameters coincide with high accuracy. The main advantage of the remote monitoring method is that the comprehensive control of a number of parameters characterizing the external state of the power system facilities, as well as its main electrical parameters, is fully automated. Specialized software for automation based on modern methods and algorithms for filtering, segmentation and image recognition is used. Also, multispectral images in the visible and infrared spectra are used, data obtained by laser location methods, as well as pictures of the electric field lines of EFs and MFs. The system contains a three-dimensional model of the power system, which is continuously updated and detailed, as well as a geographic information system (GIS), which visually allows you to represent the entire model of the system and carry out various types of control. This model also allows for effective training of UAV operators in various situations, and to develop new methods and algorithms for control and monitoring. The system contains a three-dimensional model of the power system, which is continuously updated and detailed, as well as GIS, which visually allows to represent the entire model of the system and carry out various types of control. This model also allows to conduct effective training of UAV operators in various situations, and to develop new methods and algorithms for control and monitoring [26–28]. Information received from the UAV through the communication system and from other technical monitoring systems (automatic substations and sensors) is accumulated and stored on the internal servers of the system in databases, which allows to conduct analysis and statistical processing of OPL status for any period of time. GIS allows to pave the way for maintenance crews to the accident site. Each brigade is provided with a mobile UAV complex equipped with short-range UAVs for localization and monitoring of the accident site. The functioning of the system is ensured by UAVs, which, with a given frequency and schedule, conduct multispectral surveys of power system objects, as well as measure the strength of their EF and MF. Comparison of the recorded data with the data of the standard of the research object allows to make operational decisions on its current state: to eliminate the violations that may lead to emergency failures or to find places of accidents if they could not be avoided, as well as provide places for emergencies and degradation of OPL characteristics. Using an automated system to compare the most significant parameters that describe the standard operating modes of the studied electric power facilities allows

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to prevent unauthorized power take-off from the power system. An analysis of the parameters obtained as a result of monitoring can be used to optimize both the operating modes and the structure of the investigated energy objects themselves. The main component of a remote monitoring system is an UAV. An unmanned aerial vehicle is an aircraft whose flight is under direct control of an operator located in a ground (or air) control station, using two-way communication channels or using an autopilot according to a flight mission. Recently, UAVs have been used in solving an increasing number of tasks in various fields of human activity, from space exploration and military affairs to agriculture. This is due to the presence of a number of significant advantages of this type of aircraft: lack of crew, low cost and low operating costs. Given the significant progress in the development of computer technology, its miniaturization and energy efficiency, as well as the development and practical use of new algorithms and methods, the use of UAVs can increase the efficiency of solving complex scientific and practical problems related to logistics, monitoring, control and security [29–32]. Regarding the electric power industry, UAVs allow solving the following problems [7, 33, 34]: • • • • • • • • • • •

assessment of the status of power lines; aerial photography of OPLs and power pylons; thermal imaging control of power elements of OPLs; measurement of wire sag; control of the permissible height of trees in the zone of passage of high-voltage lines; analysis of corridor overgrowth; identification of unauthorized construction sites in a protective strip; survey of new routes of OPLs and the adjacent territory, and the creation of a digital elevation model; prompt creation of a photographic plan of the construction sites of power facilities; analysis of damage, accidents; forecasting and modeling of natural impacts.

It should also be noted the problems that arise when using UAVs. Among the most significant, it is necessary to highlight: • ensuring the transmission of information through communication channels between the UAV and the control center with the necessary width and without distortion; • recognition of objects by registered information; • energy efficiency and autonomy; • safety and trouble-free. UAVs are classified according to the construction scheme [35]: • aerodynamic (airplane type: fuselage, flying wing); • aerostatic and aerostatically unloaded; • reactive;

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• helicopter and multicopter (3, 4, 6 and 8 rotary). The most common UAVs are aircraft and multicopter (helicopter) types. Each of them better solves a certain range of problems. During considering electricity monitoring issues, they are limited to these two classes, as well as unmanned balloons and tethered UAVs. Aircraft-type UAVs are used to create plans and digital terrain models and monitor extended objects. Main advantages: high speed, significant range and autonomy. Helicopter-type UAVs and multicopter are used for inspection of complex (small in length) structures or lidar surveys. Main advantages: small size, launch from any sites, the possibility of freezing. From the point of view of the tasks that can be solved, the UAVs are most suitable: • for automatic monitoring, location and mapping of OPLs, the main element of the monitoring system is an aircraft-type UAV, or combined. It is advisable to use solar panels to increase autonomy or charging systems from the mains (for combined). • for the search and localization of accident sites, as well as logistic support for repairing power lines – a helicopter-type UAV or multicopter. It is advisable to use such all-weather UAVs, because they do not require special sites for takeoff and landing, while greater autonomy is not a significant factor. • for ensuring the control and safety of power facilities—UAVs of an airplane or helicopter type or safety systems (for control of power facilities). It is advisable to use UAVs with great autonomy and all-weather to ensure all-day control.

References 1. Li, S., Li, J.: Condition monitoring and diagnosis of power equipment: review and prospective. IET 2(2), 82–91. https://doi.org/10.1049/hve.2017.0026 2. Baldi, S., Quang, T.L., Holub, O., Endel, P.: Real-time monitoring energy efficiency and performance degradation of condensing boilers. Energy Convers. Manag. 136, 329–339 (2017). https://doi.org/10.1016/j.enconman.2017.01.016 3. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Principles of construction of systems for diagnosing the energy equipment. In: Diagnostic Systems for Energy Equipments. Studies in Systems, Decision and Control, vol. 281, pp. 1–22 (2020). https://doi.org/10.1007/978-3-030-44443-3_1 4. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Technical provision of diagnostic systems. In: Diagnostic Systems for Energy Equipments. Studies in Systems, Decision and Control, vol. 281, pp. 91–133 (2020). https://doi.org/10.1007/978-3030-44443-3_4 5. Vinogradov, A., Bolshev, V., Vinogradova, A., Kudinova, T., Borodin, M., Selesneva, A.: A system for monitoring the number and duration of power outages and power quality in 0.38 kV electrical networks. In: Vasant, P., Zelinka, I., Weber, G.W. (eds.) Intelligent Computing & Optimization. ICO 2018. Advances in Intelligent Systems and Computing, vol. 866, pp. 1–10. Springer, Cham (2018) https://doi.org/10.1007/978-3-030-00979-3_1

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6. Babak, S., Babak, V., Zaporozhets, A., Sverdlova, A.: Method of statistical spline functions for solving problems of data approximation and prediction of objects state. In: CEUR Workshop Proceedings, vol. 2353, pp. 810–821 (2019). http://ceur-ws.org/Vol-2353/paper64.pdf 7. Zaporozhets, A., Kovtun, S., Dekusha, O.: System for monitoring the technical state of heating networks based on UAVs. In: Shakhovska, N., Medykovskyy, M. (eds.) Advances in Intelligent Systems and Computing IV. CCSIT 2019. Advances in Intelligent Systems and Computing, vol. 1080, pp. 935–950. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-33695-0_61 8. Zaporozhets, A., Eremenko, V., Serhiienko, R., Ivanov, S.: Methods and hardware for diagnosing thermal power equipment based on smart grid technology. In: Shakhovska, N., Medykovskyy, M. (eds.) Advances in Intelligent Systems and Computing III. CSIT 2018. Advances in Intelligent Systems and Computing, vol. 871, pp. 476–489. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-01069-0_34 9. Putrenko, V., Pashynska, N.: Risk modeling of accidents in the power system of Ukraine with using Bayesian network. In: Hu, Z., Petoukhov, S., Dychka, I., He, M. (eds) Advances in Computer Science for Engineering and Education. ICCSEEA 2018. Advances in Intelligent Systems and Computing, vol. 754, pp. 13–22. Springer, Cham (2019). https://doi.org/10.1007/ 978-3-319-91008-6_2 10. Cadini, F., Agliardi, G.L., Zio, E.: A modeling and simulation framework for the reliability/availability assessment of a power transmission grid subject to cascading failures under extreme weather conditions. Appl. Energy 185(1), 267–279 (2017). https://doi.org/10.1016/j. apenergy.2016.10.086 11. Jones, D.: Power line inspection—a UAV concept. In: 2005 The IEE Forum on Autonomous Systems (Ref. No. 2005/11271), 28 Nov. 2005, London, UK, p. 8 (2005). https://doi.org/10. 1049/ic:20050472 12. Hou, J.P., Li, R., Wu, X.M., Yu, H.Y., Zhang, Z.J., Chen, Q.Y., Wang, Q., Li, X.W., Zhang, Z.F.: Microstructure evolution and strength degradation mechanisms of high-strength Al–Fe wire. J. Mater. Sci. 54, 5032–5043 (2019). https://doi.org/10.1007/s10853-018-3060-3 13. Skarbek, L., Zak, A., Ambroziak, D.: Damage detection strategies in structural health monitoring of overhead power transmission system. In: 7th European Workshop on Structural Health Monitoring, July 8–11, 2014, La Cité, Nantes, France, pp. 663–670 (2014) 14. Li, L.: The UAV intelligent inspection of transmission lines. In: International Conference on Advances in Mechanical Engineering and Industrial Informatics, pp. 1542–1545 (2015) 15. Adabo, G.J.: Unmanned aircraft system for high voltage power transmission lines of Brazilian electrical system. In: AUVSI’s Unmanned Systems 2013. Washington, USA (2013) 16. Tian, F., Wang, Y., Zhu, L.: Power line recognition and tracking method for UAVs inspection. In: 2015 IEEE International Conference on Information and Automation, 8–10 Aug. 2015, Lijiang, China, pp. 2136–2141. https://doi.org/10.1109/ICInfA.2015.7279641 17. Karakose, E.: Performance evaluation of electrical transmission line detection and tracking algorithms based on image processing using UAV. In: 2017 International Artificial Intelligence and Data Processing Symposium (IDAP), 16–17 Sept. 2017, Malatya, Turkey, pp. 1–5. https:// doi.org/10.1109/IDAP.2017.8090302 18. Toth, J., Glipin-Jackson, A.: Smart view for a smart grid—unmanned aerial vehicles for transmission lines. In: 2010 1st International Conference on Applied Robotics for the Power Industry, 5–7 Oct. 2010, Montreal, QC, Canada, pp. 1–6. https://doi.org/10.1109/CARPI.2010.5624465 19. Zhang, F., Wang, W., Zhao, Y., Li, P., Lin, Q., Jiang, L.: Automatic diagnosis system of transmission line abnormalities and defects based on UAV. In: 2016 4th International Conference on Applied Robotics for the Power Industry (CARPI), 11–13 Oct. 2016, Jinan, China, pp. 1–5. https://doi.org/10.1109/CARPI.2016.7745632 20. Cao, W., Zhu, L., Han, J., Wang, T., Du, Y.: High voltage transmission line detection for uav based routing inspection. In: 2013 IEEE/ASME International Conference on Advanced Intelligent Mechatronics, 9–12 July 2013, Wollongong, NSW, Australia. https://doi.org/10. 1109/AIM.2013.6584150

8

M. M. Rezinkina et al.

21. Li, Z., Mu, S., Li, J., Wang, W., Liu, Y.: Transmission line intelligent inspection central control and mass data processing system and application based on UAV. In: 2016 4th International Conference on Applied Robotics for the Power Industry (CARPI), 11–13 Oct. 2016, Jinan, China, pp. 1–5. https://doi.org/10.1109/CARPI.2016.7745631 22. Wu, Y., Zhao, G., Hu, J., Ouyang, Y., Wang, S.X., He, J., Gao, F., Wang, S.: Overhead transmission line parameter reconstruction for UAV inspection based on tunneling magnetoresistive sensors and inverse models. IEEE Trans. Power Delivery 34(3), 819–827 (2019). https://doi. org/10.1109/TPWRD.2019.2891119 23. Zhang, Y., Yuan, X., Hang, Y., Chen, S.: UAV low altitude photogrammetry for power line inspection. Int. J. Geo-Inf. 6(1), 14 (2017). https://doi.org/10.3390/ijgi6010014 24. Wang, Q., Xiong, H., Qiu, B.: The attitude control of transmission line fault inspection UAV based on ADRC. In: 2017 International Conference on Industrial Informatics—Computing Technology, Intelligent Technology, Industrial Information Integration (ICIICII), 2–3 Dec. 2017, Wuhan, China, pp. 186–189. https://doi.org/10.1109/ICIICII.2017.48 25. He, T., Zeng, Y., Hu, Z.: Research of multi-rotor UAVs detailed autonomous inspection technology of transmission lines based on route planning. IEEE Access 7, 114955–114965 (2019). https://doi.org/10.1109/ACCESS.2019.2935551 26. Pereira, F.C., Pereira, C.E.: Embedded image processing systems for automatic recognition of cracks using UAVs. IFAC-PapersOnLine 48(10), 16–21 (2015). https://doi.org/10.1016/j.ifa col.2015.08.101 27. Zhang, Y., Yuan, X., Li, W., Chen, S.: Automatic power line inspection using UAV images. Remote Sensing 9(8), 824 (2017). https://doi.org/10.3390/rs9080824 28. Montambault, S., Beaudry, J., Toussaint, K., Pouliot, N.: On the application of VTOL UAVs to the inspection of power utility assets. In: 2010 1st International Conference on Applied Robotics for the Power Industry, 5–7 Oct. 2010, Montreal, QC, Canada, pp. 1–7. https://doi. org/10.1109/CARPI.2010.5624443 29. Zhou, G., Yuan, J., Yen, I.-L., Bastani, F.: Robust real-time UAV based power line detection and tracking. In: 2016 IEEE International Conference on Image Processing (ICIP), 25–28 Sept. 2016, Phoenix, AZ, USA, pp. 744–748. https://doi.org/10.1109/ICIP.2016.7532456 30. Hui, X., Bian, J., Zhao, X., Tan, M.: Deep-learning-based autonomous navigation approach for UAV transmission line inspection. In: 2018 Tenth International Conference on Advanced Computational Intelligence (ICACI), 29–31 March 2018, Xiamen, China, pp. 455–460. https:// doi.org/10.1109/icaci.2018.8377502 31. Deng, C., Wang, S., Huang, Z., Tan, Z., Liu, J.: Unmanned aerial vehicles for power line inspection: a cooperative way in platforms and communications. J. Commun. 9(9), 687–692 (2014) 32. Bian, J., Hui, X., Zhao, X., Tan, M.: A novel monocular-based navigation approach for UAV autonomous transmission-line inspection. In: 2018 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 1–5 Oct. 2018, Madrid, Spain, pp. 1–7. https://doi. org/10.1109/IROS.2018.8593926 33. Zaporozhets, A.: System for diagnosing main pipelines of heat networks based on UAVs. Int. J. “NDT Days” 2(1), 69–77 (2019) 34. Boukoberine, M.N., Zhou, Z., Benbouzid, M.: A critical review on unmanned aerial vehicles power supply and energy management: solutions, strategies, and prospects. Appl. Energy 255, 113823 (2019). https://doi.org/10.1016/j.apenergy.2019.113823 35. Zaporozhets, A.: Overview of quadrocopters for energy and ecological monitoring. In: Babak, V., Isaienko, V., Zaporozhets, A. (eds.) Systems, Decision and Control in Energy I. Studies in Systems, Decision and Control, vol. 298, pp. 15–36 (2020) https://doi.org/10.1007/978-3-03048583-2_2

Improving Methods for One-Sided Determination of the Location of Damage of Overhead Power Lines in Networks with Effectively Grounded Neutral Based on UAVs Gennadiy A. Senderovich , Artur O. Zaporozhets , Oleg G. Gryb , Ihor T. Karpaliuk , Sergiy V. Shvets , and Natalia V. Rudevich Abstract Methods for one-sided determination of the location of power lines’ damage are of interest to all parts of the companies involved in the supply and redistribution of electricity. The advantage of one-sided methods for determining the location of damage by emergency mode parameters is the ability to determine the distance to the place of damage without transmitting information about emergency mode parameters from the opposite end of the line. Such methods are extremely promising from the point of view of reducing the costs of maintaining power lines. The section provides a classification of methods for one-sided determination of the location of damage. The mathematical model developed by the authors is described for one-sided determination of the place of damage to the power line by emergency mode parameters for the cases of single-phase, two-phase, three-phase short circuits. A technique for determining the location of damage of a line is disclosed. Keywords Location of damage · Power lines · Emergency mode parameters · Short circuit · Mathematical model G. A. Senderovich · O. G. Gryb · I. T. Karpaliuk · S. V. Shvets · N. V. Rudevich Department of Automation and Cybersecurity, National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine e-mail: [email protected] O. G. Gryb e-mail: [email protected] I. T. Karpaliuk e-mail: [email protected] S. V. Shvets e-mail: [email protected] N. V. Rudevich e-mail: [email protected] A. O. Zaporozhets (B) Department of Monitoring and Optimization of Thermophysical Processes, Institute of Engineering Thermophysics of NAS of Ukraine, Kyiv, Ukraine e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 Y. I. Sokol and A. O. Zaporozhets (eds.), Control of Overhead Power Lines with Unmanned Aerial Vehicles (UAVs), Studies in Systems, Decision and Control 359, https://doi.org/10.1007/978-3-030-69752-5_2

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Improving the accuracy of one-sided LoD of OPLs is traditionally of interest to energy enterprises due to the interest of operating personnel in reducing time and costs for LoD. Violation of the normal mode of operation of electric power systems, as a rule, occurs due to damage to its elements, in OPLs. This leads to disruption of power supply, increased losses and reduced quality of electric energy, and, as a result, negatively affects on the technical and economic performance of both consumers and suppliers of electric energy [1–5]. A speedy return to the normal mode of operation of electrical systems, in cases of damage of OPLs, depends on the fast and accurate of LoD. The advantage of one-way LoD methods according to the EMP is the ability to determine the distance to the place of damage without transmitting information about the EMP from the opposite end of the line. Using one-way methods eliminates the need for synchronous information from both sides of the OPL. This significantly reduces the cost of the equipment used, allows to measure the distance to the place of damage in the absence of synchronous communication channels, increases the reliability and safety of operation of electric power systems [6–8]. The most widespread are one-sided methods of LoD, based on the determination of the resistance of the SC loop, which can be used both graphically and analytically [9–11]. The main and significant disadvantage is the deviation of the calculated distances, which depends on the unknown value of the transition resistance at the damage location Rd , which is understood as the total resistance of the circuit along which the SC flows between the phases during interfacial faults or between phase and ground during SC to ground. With interphase SCs, the main component of the transition resistance is the resistance of the electric arc Ra , and with single-phase short-circuit Rd is additionally determined by the ground resistance of the support. In addition to the transient resistance, the accuracy of one-sided LoD is affected by branch currents and magnetically coupled lines. The use of digital emergency mode recorders made on the basis of a computer makes it possible to record the instantaneous values of EMP in the interval of the emergency mode, to allocate the time of a stable SC, to determine the interrelated complex values of the parameters of the stable SC mode. There is no need to work in real time, so it can be used almost any algorithms for processing registrar records [12]. Modern tools increase the speed and accuracy of EMP, but do not exclude the need for a visual study of the consequences of emergency shutdowns and routine inspections of OPLs. An important and most economical way to conduct these activities may be the use of UAVs [13–15].

Improving Methods for One-Sided Determination …

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1 Mathematical Model of One-Sided Determination of the LoD of OPL by EMP Let consider a mathematical model of a one-sided LoD based on measuring the loop resistance using a EMP line with a grounded neutral. To compile equations describing the state of the model in cases of one-, two- and three-phase SCs, it can be applied the method of symmetrical components. The digital recorder of emergency modes based on electronic computers allows using instantaneous values of currents and voltages of a three-phase network recorded for 1–3 periods of the stable SC mode to obtain complex values of the effective values of currents, voltages, as well as their symmetrical components [16]. In the preparation of the equations, the following factors were taken into account that affect the accuracy of the LoD: • transition resistance of damage site; • branch line; • magnetically connected lines. More about emergency mode of electric power equipment is shown in [17, 18].

1.1 Single-Phase SC Next, the general case of a single-phase SC through a transition resistance in a doublestranded line with a branch with two-way power is considered [19–21]. The network equivalent circuit for the considering damage (Fig. 1) provides for the installation of a DR from the C1 system.

Fig. 1 Network equivalent circuit for the general case of single-phase SC

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The equation drawn up on the SC loop for the damaged phase can be written as follows [22]: I ab I ab I I ab I bc I bc · Z ab · Z ab · Z ab · Z bc · Z bc U ϕ = I 1I ab · Z ab 1 + I2 2 + I0 0 + I0 M + I1 1 + I2 2   I I bc bc I bc I I bc C2 I , (1) + I 0I bc · Z bc + I · Z + 3 · R + I + I d 0 0 0 M 0 0

where I I1ab , I I2ab , I I0ab , I I1ab , I I2bc , I I0bc —currents of the forward, reverse and zero sequences of the damaged phase of OPL №1 in sections ab and bc, respectively; ab ab bc bc bc Z ab 1 , Z 2 , Z 0 , Z 1 , Z 2 , Z 0 —resistance of the forward, reverse and zero sequences of the sections ab and bc; I II0 ab , I II0 bc —zero sequence currents of a parallel bc OPL (№ 2) in sections ab and bc; Z ab M , Z M —mutual resistance of lines in the section C2 ab and bc; I 0 —zero sequence current of SC from system C2 from the opposite end of the line. According to the classical representation of a single-phase SC to ground, the currents of the symmetrical components in the SC location are equal: I 1 = I 2 = I 0.

(2)

Assuming that condition (2) applies to the SC supply branches, Eq. (1) takes the form I I ab I bc I I bc · Z ab · Z bc · Z bc Uϕ = I 0I ab · Z ab Σ + I0 M + I0 Σ + I0 M + 3Rd  I bc  I I bc C2 · I0 + I0 + I0

(3)

bc where Z ab Σ , Z Σ are the total resistances of the forward, reverse and zero sequences of the sections ab and bc, defined as ab ab ab bc bc bc bc Z ab Σ = Z1 + Z2 + Z0 ; ZΣ = Z1 + Z2 + Z0 .

In the general case, equality (2) is only approximately satisfied for the supply branches. The main reason for this is the presence of load currents. For a correct description, additives were introduced into expression (3):   I ab − 3 · I 0I ab · Z ab ΔI ϕI ab · Z ab 1 = Iϕ 1 ;  bc  I bc I bc bc I bc · Z1 , ΔI ϕ · Z 1 = I ϕ − 3 · I 0

(4)

where I Iphab and I Iphbc are the short-circuit phase currents in sections ab and bc of damaged line № 1. The introduced additives take into account the real ratio of the currents of symmetrical components. The final equation for the SC loop will be as follows:

Improving Methods for One-Sided Determination … I bc I I bc Uϕ = ΔU 1 + I 0I bc · Z bc · Z bc · Z bc Σ + ΔI ϕ 1 + I0 M + ΔU R ,

13

(5)

where ΔU 1 —voltage drop in section ab, defined as I I ab I ab ab · Z ab ΔU 1 = I 0I ab · Z ab Σ + I0 M + ΔI ϕ · Z 1 .

(6)

ΔU R is a voltage drop across the transition resistance, defined as   = 3 · Rd · I 0K . ΔU R = 3 · R · I 0I bc + I 0I I bc + I C2 0

(7)

Complex equation (5) contains three unknown quantities: the transition resistance bc Rd , the zero sequence current I C2 0 from the other end of the line and the resistance Z , due to the distance to the damage location. Given that complex numbers are considered equal to each other, if their real and imaginary parts are separately equal, the complex equation (5) is written below in the form of a system of two real equations:     I I bc  bc ⎫    · rM ⎪ Re Uϕ − Re(ΔU1 ) = Re I0I bc · rΣbc − Im I0I bc · xΣbc + Re ⎪  I0 ⎪    bc  ⎪ ⎪ I bc ⎪ · x1bc −Im I0I I bc · x M + Re ΔIϕI bc · r1bc − Im ΔI ph ⎪ ⎪ ⎬ +Re(ΔU ), R (8)  I bc  bc  I I bc  bc  I bc  bc   ⎪ · x · r · x + Im I + Re I = Re I Im Uϕ − Im(ΔU ) ⎪ 1 Σ Σ 0 0 0 M ⎪  bc      ⎪ ⎪ ⎪ +Im I0I I bc · r M + Re ΔIϕI bc · x1bc + Im ΔIϕI bc · r1bc ⎪ ⎪ ⎭ +Im(ΔU R ), where    ab    Re(ΔU1 ) = Re I0I ab · rΣab − Im I0I ab · xΣab + Re I0II ab · r M  ab      − Im I0II ab · x M + Re ΔIϕI ab · r1ab + Im ΔIϕI ab · x1ab , and       ab − Im I0I ab · rΣab + Re I0II ab · X ab Im(ΔU1 ) = Re I0I ab · X Σ M  ab      + Im I0II ab · r M + Re ΔIϕI ab · X 1ab + Im ΔIϕI ab · r1ab . In the system of equations (8), all resistances are expressed using coupling coefficients through the reactance of the direct sequence x 1 : rΣ = K R · x1 ; xΣ = K X · x1 ; r M = K MR · x1 ; x M = K MX · x1 ; r1 = K R1 · x1 . The desired distance bc is part of the line bd. Taking into account the equality  bc bd of the resistivities of sections bc and bd Z sp = Z sp , the coupling coefficients are determined from the known parameters of the part bd:

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KR =

G. A. Senderovich et al. bd bd 2 · r1bd + r0bd x0bd rM xM r1bd ; K = 2 + ; K = ; K = ; K = . (9) X MR MX R1 x1bd x1bd x1bd x1bd x1bd

Taking into account the coupling coefficients in the system of equations (8), it can be expressed the resistances through the reactance of the direct sequence of section bd: ⎫  I bc   I I bc   I bc    · K · K · K − Im I + Re I = [Re I Re Uϕ − Re(ΔU ⎪ ) 1 R X M R 0 0 0 ⎪       ⎪ ⎪ ⎪ −Im I0I I bc · K M X + Re ΔIϕI bc · K R1 − Im ΔIϕI bc ] · x1bc ⎪ ⎪ ⎬ +Re(ΔU ), R      I bc    I bc I I bc · K X + Im I0  · K R+ Re I0 · KMX ⎪ Im Uϕ − Im(ΔU ⎪  I I1bc) = [Re I0 ⎪ ⎪ bc I bc I bc ⎪ +Im I0 · K M R + Re ΔIϕ + Im ΔIϕ · K R1 ] · x1 ⎪ ⎪ ⎭ +Im(ΔU R ). (10) The equations of system (10) take into account the real and imaginary components of the voltage drop in the transition resistance Rd , based on expression (7), and each equation of system (10) is solved with respect to Rd : Re(Uϕ )−Re(ΔU1 )−Re(IΣ )·x1bc Re(I0K ) Im(Uϕ )−Im(ΔU1 )−Im(IΣ )·x1bc Im(I0K )

= 3 · Rd ;



= 3 · Rd ,

(11)

where       Re(IΣ ) = Re I0I bc · K R − Im I0I bc · K X + Re I0I I bc · KMR       − Im I0I I bc · KMX + Re ΔIϕI bc · K R1 − Im ΔIϕI bc ;       Im(IΣ ) = Re I0I bc · K R + Im I0I bc · K R + Re I0I I bc · KMX       + Im I0I I bc · KMR + Re ΔIϕI bc + Im ΔIϕI bc · K R1 ;         Re I 0K = Re I I0bc + Re I II0 bc + Re I C2 0 ;  II bc   C2   I bc  Im(I0K ) = Im I0 + Im I0 + Im I0 . The resulting system of equations (11) allows to exclude an unknown quantity Rd :     Re Uϕ − Re(ΔU1 ) − Re(IΣ ) · x1bc Im Uϕ − Im(ΔU1 ) − Im(IΣ ) · x1bc = . Re(I0K ) Im(I0K ) (12) The solution of equation (12) allows to find the reactance of the direct sequence in an unknown area bc:

Improving Methods for One-Sided Determination …

x1bc

         Re Uϕ − Re ΔU 1 − Im Uϕ − Im ΔU 1 · =   Re I   Re I Σ − Im I Σ · Im( I 0K ) ( 0K )

15 Re( I 0K ) Im( I 0K )

.

(13)

Formula (13) is not completely defined. It contains an unknown value of the zero sequence current I C2 0 from the C2 system as part of the current I 0K . This is obvious if (13) is presented in a different form:

x1bc

         Re Uϕ − Re ΔU 1 − Im Uϕ − Im ΔU 1 · ctgϕ R     = , Re I Σ − Im I Σ · ctgϕ R

(14)

where         Re I 0K Re I 0I bc + Re I 0I I bc + Re I C2 =      0   ctgϕ R = Im I 0K Im I 0I bc + Im I 0I I bc + Im I C2 0

(15)

SC currents in the network of one degree of voltage are close in phase. In particular, for a 110 kV network, the arguments of the complex values of the SC currents are approximately 70°. It can be made an assumption about the equality of the phases of the zero sequence currents, which are components of the current I 0K . With this assumption, expression (14) is defined, since ctgϕ R can be calculated by the formula, which does not include I C2 0 :     Re I 0I bc + Re I 0I I bc    . ctgϕ R ≈ Im I 0I bc + Im I 0I I bc

(16)

It should be noted that for the line with one-sided power supply, formula (14) is also determined mathematically correct, since in this case I C2 0 = 0. Based on the above calculations, it is possible to propose a calculated expression for determining the distance to the place of damage during a single-phase SC according to the EMP of one of the supply ends of the line in a 110 kV network. The formula for the case under consideration can be represented in general form:         Re U ϕ − Re ΔU 1 − Im U ϕ − Im ΔU 1 · ctgϕ R 1     · bd , (17) L X = L ab + x1 sp Re I Σ − Im I Σ · ctgϕ R where L x —required distance; L ab —length of line to branch; x1bdsp —specific reactance of the direct sequence of the line segment bd. Depending on the location of the SC point and the electrical connections of the power line, the application of the obtained formula has its own characteristics.

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For calculating L x at a single-phase fault which is located on a double-chain line after a branch (Fig. 1), formula (17) should be used in its entirety with the definition of its components using expressions (8–16). In the case of a single-phase SC on a double-chain line, before the branch or in the absence of a branch in formula (17) it should be taken L ab = 0; Re(ΔU1 ) = 0; Im(ΔU1 ) = 0. Formula (17) is also intended for calculating single-chain lines. In this case, during determining the real and imaginary components of the currents, the components due to magnetic coupling with a parallel connected line are not taken into account. The corresponding components in the calculated expressions for the stresses disappear. Branch influence. The presence of a branch on the damaged line during SC behind the branch changes the flow distribution along the entire line [23–25]. With using the recorder at one end of the line (Fig. 1), the presence of a branch leads to the appearance of an additional unknown quantity in the expressions defining the components of formula (13). This value is I Iϕbc —the current of the damaged line after the branch. The difference between the currents at the measurement location I Iϕab and the current after the branch I Iϕbc is determined by the branch current: I ϕI ab + I ϕI r es = I ϕI bc .

(18)

With taking into account the branch, it should be noted that the branch current affects the mode of operation of branch transformers. The presence of grounded neutrals of branch transformers leads to the appearance of a current of a zero branch sequence, which mainly determines the branch current in a single-phase SC. For taking into account the current of the zero branch sequence, the case of a singlephase SC on a double-chain line with a branch in the closed mode of operation of the neutrals of branch transformers is considered. According to the equivalent circuit of the zero sequence of the network for the considered damage (Fig. 2), the zero sequence voltage at the connection point of the branch is: U 0I r es = U 0I − ΔU 0I ab ; U 0I I r es = U 0I I − ΔU 0I I ab ,

(19)

where U 0I and U II0 are zero sequence voltage on the OPL buses from the measurement side, to which lines №1 and №2 are connected; ΔU I0ab and ΔU II0 ab are voltage drop of the zero sequence in the section ab of lines №1 and №2, which is due to the following expressions: I I ab I I ab I ab · Z ab = I 0I I ab · Z ab · Z ab ΔU 0I ab = I 0I ab · Z ab 0 + I0 M ; ΔU 0 0 + I0 M.

(20)

On the other hand, the voltage at the junction of the branches can be determined through the currents of the zero sequence of branches:   U I0res = − I I0res · Z I0res + I II0 res · Z res M ;

Improving Methods for One-Sided Determination …

17

Fig. 2 Network equivalent circuit of the zero sequence of the network in the case of a single-phase SC on a double-circuit line with a branch

  II res U II0 res = − I I0res · Z res · Z II0 res , M + I0

(21)

where Z I0res and Z II0 res —the total resistance of the zero sequence of the branches that are connected to lines №1 and №2, and consisting of the resistance of the zero sequence of the line and transformers: Z I0res = Z I0lin + Z I0tr , Z II0 res = Z II0 lin + Z II0 tr ; Z res M —resistance of mutual induction of the branch lines №1 and №2. The joint solution of equations (19) and (21), taking into account expression (20), gives the values of the currents of the zero branch sequence:

I 0I r es

  I r es    −U I r es Z r es  Z −U 0I r es  M  0   0  −U I I r es Z I I r es   Z r es −U I I r es  0 M 0 I I r es ; I =  I0r es =   I r es  . 0 r es  r es  Z Z Z Z M  M   0  0  Z r es Z I I r es   Z r es Z I I r es  M 0 M 0

(22)

Let consider accounting the currents of the forward and reverse branch sequences. The currents of the forward and reverse branch sequences will occur under any operating conditions of the neutrals of branch transformers, since their flow path is not connected to the ground. Their accurate calculation is difficult with taking into account the real load resistance and possible recharge from the engines. With known load resistances and neglecting recharge from the motors, the currents of the forward and reverse sequences can be calculated using an equivalent circuit, the difference from the zero sequence equivalent circuit is that there is no resistance to the relationship between parallel lines (Fig. 3). Similarly to the zero sequence of the voltage of the forward and reverse sequences at the connection point of the branch, they are determined, on the one hand, as

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Fig. 3 Equivalent circuit of the forward and reverse network sequences in the case of a single-chain SC on a double-chain line behind a branch

U 1I r es = U 1I − I 1I ab · Z 1I ab ; U 2I r es = U 2I − I 2I ab · Z 2I ab ,

(23)

on the other hand, through the currents of the branches of the forward and reverse sequences U 1I r es = −I 1I r es · Z 1I r es ; U 2I r es = −I 2I r es · Z 2I r es ,

(24)

where U 1I and U 2I —voltage of the forward and reverse sequences at the measurement point; Z I1res and Z I2res —total resistance of the forward and reverse branch sequences: Z I1res = Z I1lin + Z I1tr + Z I1load , Z I2res = Z I2lin + Z I2tr + Z I2load . The joint solution of equations (23) and (24) allows to determine the desired branch currents in this form: I 1I r es =

−U 1I r es I r es −U 2I r es I = . 2 Z 1I r es Z 2I r es

(25)

The total branch current in the damaged phase is: I ϕI r es = I 1I r es + I 2I r es + I 0I r es .

(26)

Based on the calculations made in formula (17), it is necessary to take into account the change in currents by branching in section bc (Fig. 3):             Re I I0bc = Re I I0ab + Re I I0res ; Im I I0bc = Im I I0ab + Im I I0res ;             Re I II0 bc = Re I II0 ab + Re I II0 res ; Im I II0 bc = Im I II0 ab + Im I II0 res , as well as the current component that takes into account the inequality of the currents of symmetrical components in accordance with (4):

Improving Methods for One-Sided Determination …

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      Re ΔI Iϕbc = Re ΔI Iϕab + Re ΔI Iϕres        = Re I Iϕab − Re 3 · I I0ab + Re I Iϕres − Re 3 · I I0res ;       Im ΔI I bc = Im ΔI Iϕab + Im ΔI Iϕres       I ab  ϕ  = Im I ϕ − Im 3 · I I0ab + Im I Iϕres − Im 3 · I I0res , 

where ΔI Iφres = I Iφres − 3 · I I0res .

1.2 Two-Phase SC Let consider the case of a two-phase SC through a transition resistance [26, 27]. Suppose that a SC occurred due to an arc between phases B and C. The equivalent circuit of the network of the considered damage is shown in Fig. 4. The equation drawn up on the SC loop for the damaged phases is written as follows:   U B − U C = I B − I C · Z + ΔU R ,

(27)

where Z is a resistance of the desired line segment. The voltage drop across the transition resistance is determined by the following expression:   C2 · Rd , ΔU R = I C1 R + IR

(28)

C2 where I C1 R and I R —SC currents from systems C1 and C2, which flow through a transition resistance. It should be noted that the load current I load (Fig. 4) will not pass through the transition resistance, because:

Fig. 4 Network equivalent circuit for two-phase SC through transition resistance

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G. A. Senderovich et al.

C2 C2 C2 I C1 R + I R = I B − IC + I B − IC       C2 C2 = I B1 + I B2 + I B load − I C1 + I C2 + I C load + I C2 B1 + I B2 − I B load       C2 C2 − I C2 C1 + I C2 − I C load = I B1 + I B2 − I C1 + I C2    C2  C2 C2 + I C2 (29) B1 + +I B2 − I C1 + I C2 ,

C2 C2 C2 where I B1 , I B2 , I C1 , I C2 , I C2 B1 , I B2 , I C1 , I C2 —direct and reverse sequence currents of damaged phases from system C1 and system C2, respectively; I B load and I C load — load currents of damaged phases. Equation (27) are presented below in the following form:

U = I · Z + I R · Rd ,

(30)

where U = U B − U C —voltage difference of damaged phases; I = I B − I C — C2 currents difference of damaged phases; I R = I C1 R + I R —SC current through transition resistance. In the form of a system of two real equations, Eq. (30) has the form 

        Re U  = Re  I  · r − Im I  · x + Re  I R  · Rd ; Im U = Re I · x + Im I · r + Im I R · Rd .

(31)

The system of equations (31) can be written in terms of the coupling coefficient K R = r/ X : 

Re(U ) = [Re(I ) · K R − Im(I )] · x + Re(I R ) · Rd ; Im(U ) = [Re(I ) + Im(I ) · K R ] · x + Im(I R ) · Rd .

(32)

In order to exclude the value of Rd from Eq. (32), it can be solved with respect to the reactance of the desired section of the line x: x=

Re(U ) − Im(U ) · ctgϕ R , Re(I ) · (K R − ctgϕ R ) − Im(I ) · (1 + K R · ctgϕ R )

(33)

R) where ctgϕ R = Re(I —cotangent of the SC current triangle. Im(I R ) In formula (33), all components are known, except for the value ctgϕ R , for which information is not sufficiently defined. It can be written ctgϕ R , taking into account (29), as follows:

    Re(I B ) − Re(IC ) + Re I BC2 − Re ICC2   .  ctgϕ R = Im(I B ) − Im(IC ) + Im I BC2 − Im ICC2

(34)

According to the classical representation of a two-phase SC, the current in a special phase (in our case, phase A) at the SC location is equal to zero (Fig. 5):

Improving Methods for One-Sided Determination …

21

Fig. 5 Symmetric components of the currents through the arc with a two-phase SC

I R A = I R A1 + I R A2 = 0.

(35)

Next, it be will be taken into account the relationship between the symmetrical components of the currents of the two-phase SC of various phases (Fig. 5). In this case, the total current of the two-phase short circuit passing through the arc is √     I R = I R1B + I R2B − I R1C + I R2C = 2 · 3·I R2 A ∠(ϕ A2 + 90).

(36)

With (36), Eq. (34) takes the form: ctgϕ R =

− sin ϕ A2 I R · cos ϕ R cos(ϕ A2 + 90◦ ) = = = −tgϕA2 . I R · sin ϕ R sin(ϕ A2 + 90◦ ) cos ϕ A2

(37)

Taking into account (37), Eq. (33) can be written as follows:     Re U + Im U · tgϕ A2     , X= Re I · (K R + tgϕ A2 ) + Im I · (K R · tgϕ A2 − 1)

(38)

where tgϕA2

 C2  Im(I A2 ) − Im I A2  C2  , = Re(I A2 ) − Re I A2

(39)

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    Re I A2 , Im I A2 —the real and imaginary components of the current in the    C2  reverse sequence of the special phase from the C1 system; Re I C2 A2 and Im I A2 — the real and imaginary components of the current of the reverse sequence of the special phase from the C2 system.   C2 Assuming that I A2 and I C2 A2 have similar phase values ϕA2 ≈ ϕA2 and take into account that the load currents do not affect calculation (29), formula (38) is well defined during measurements on one side of the line, while tgϕA2

  Im I A2  . ≈ Re I A2

(40)

Based on the calculations made, it is possible to propose a calculated expression for determining the distance to the place of damage during a two-phase SC according to the EMP of one of the supply ends of the line in networks with effectively grounded neutral     Re U + Im U · tgϕ A2 1     · , (41) LX = Re I · (K R + tgϕ A2 ) + Im I · (K R · tgϕ A2 − 1) xsp where x sp —specific reactance of a line; tgϕA2 —reverse phase tangent in a particular phase.

1.3 Three-Phase SC Let consider the case of a three-phase SC through a transition resistance [28–30]. The Network equivalent circuit of the damage in question is shown in Fig. 6. The equation made up of a SC loop for single-phase can be written as   · Rd , U ϕ = I ϕ · Z + I ϕ + I C2 ϕ

(42)

Fig. 6 Network equivalent circuit in the case of a three-phase SC through transition resistance

Improving Methods for One-Sided Determination …

23

where I ϕ —SC phase current from the measurement side; I C2 φ —SC phase current from the C2 system. Equation (42) in the form of a system of two real equations has the form 

     

    Re U ϕ = Re I ϕ · r − Im I ϕ · x + Re I ϕ + Re I C2 ·R ;    

   ϕ  d   · Rd . Im U ϕ = Im I ϕ · x + Re I ϕ · r + Im I ϕ + Im I C2 ϕ

(43)

With the coupling coefficient K R = r/x, Eq. (43) can be written as: 

        Re U ϕ = Re I ϕ · K R − Im I ϕ · x + Re I K ] · Rd ;          Im U ϕ = Im I ϕ + Re I ϕ · K R · x + Im I K ] · Rd ,

(44)

        where Re(I K ) = Re Iϕ + Re IϕC2 ; Im(I K ) = Im Iϕ + Im IϕC2 . The solution of Eq. (44) for the reactance of the desired section of line x can be represented as follows:     Re U ϕ − Im U ϕ · ctgϕκ     , x= Re I ϕ · (K R − ctgϕκ ) − Im I ϕ · (1 + K R · ctgϕκ )

(45)

      Re I ϕ + Re I C2 Re I K ϕ  =   ctgϕ K = . Im I K Im I ϕ + Im(I C2 ϕ )

(46)

where

Assuming the equality of the phases of the currents I ϕ and I C2 ϕ ctgϕ K ≈

Re(I ) . Im(I )

(47)

For assumption (47), Eq. (45) is transformed as follows:     Re I Re Uϕ − Im Uϕ · Im( Iϕ ) ( ϕ) x=       Re( Iϕ ) − Im Iϕ · 1 + K R · Re Iϕ · K R − Im I ( ϕ)         Re Uϕ · Im Iϕ − Im Uϕ · Re Iϕ . =  2 Iϕ

Re( Iϕ ) Im( Iϕ )



(48)

If the real axis is selected, which coincides in direction with U ϕ , then Eq. (48) can be simplified:

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G. A. Senderovich et al.

x =−

Uϕ · sinα , Iϕ

(49)

where α—current argument I ϕ . If for the inductive circuit α = −βϕ , where βϕ is the angle between the voltage and current of the same phase, then expression (49) can be written as x=

U · sin βϕ . Iϕ

(50)

The calculated expression for determining the distance to the place of damage during a three-phase SC according to the EMP of one of the supply ends of the line in networks with an effectively grounded neutral has the form Lx =

U · sin βϕ . Iϕ · xsp

(51)

2 Methodology for Determining the LoD of OPL by Mode Parameters Obtained on One of Its Sides The expressions obtained in paragraph 1 allows to do a one-way LoD in EMP of OPL in networks with effectively grounded neutral. However, the accuracy of the results obtained with bilateral power supply to the damaged line will depend on the influence of the C2 system on the opposite ends of the line. One way to account for this effect is to use the parameters of the network equivalent circuit relative to OPL, which is proposed to be determined using equivalent programs [31]. This approach to accounting for the C2 system has high accuracy, but there is a disadvantage in linking to equivalent programs that require constant correction during the operation of networks. Another way to account for the C2 system is to use its parameters determined using digital emergency mode recorders, which are based on a computer [32]. The DR make it possible to record the instantaneous EMP values in the interval of emergency conditions during SC that are out of line from the opposite side of the protected zone. The parameters C2 obtained by calculation can be stored in the computer memory until the moment of SC on the line, and then used to refine the calculation of the LoD with the date and time of the state of the C2 system. The parameters of the C2 system act as a support for the forward, reverse, and zero sequences, as well as the direct sequence of EMF.

Improving Methods for One-Sided Determination …

25

2.1 Calculation of System Parameters of the Opposite End of the Line In case of damages in the C1 system, SC currents from the C2 system pass through the line controlled by the DR (Fig. 7). Information about the SC currents recorded by the DR during damage to the C1 system allows to calculate the EMP inherent in the C2 system at the time of the SC. Next, the calculation of the EMP at installing the DR on a single-chain line is considered. The equivalent circuit of the zero and reverse sequences of a single-chain line differ only quantitatively (Fig. 8). According to the Kirchhoff’s second law, it can be mae an equation for the zero and inverse sequences: U ϕ0 = I 0 · Z C2 0 + I 0 · Z 0L ,

(52)

U ϕ 2 = I 2 · Z C2 2 + I 2 · Z 2L .

(53)

From Eqs. (52) and (53), the resistances of the zero and reverse sequences of the system C2 are determined: Z C2 0 =

U ϕ0 I0

− Z 0,

(54)

Fig. 7 SC inside the C1 system (design scheme)

Fig. 8 Equivalent circuit of the zero (reverse) network sequence at installing the DR on a singlechain line

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Fig. 9 Equivalent circuit of the direct network sequence at installing a DR on a single-chain line

Z C2 2 =

U ϕ2 I2

− Z 2.

(55)

The network equivalent circuit of direct sequence is characterized by the presence of EMF E C2 in the C2 system (Fig. 9). The equations according to Kirchhoff’s second law for a direct sequence will have the form   U ϕ1 = E C2 + I 1 · Z 1 + Z C2 1 .

(56)

Equation (56) can be approximately solved with respect to E C2 under the C2 assumption that Z C2 1 ≈ Z2 :   E C2 ≈ U ϕ1 − I 1 · Z 1 + Z C2 2 .

(57)

Let consider the calculation of EMP at installing a DR on a double-chain line (Fig. 10). In the equation compiled according to Kirchhoff’s second law for the zero sequence, it is necessary to take into account the magnetic coupling with the parallel line:   U ϕ0 = I 0I · Z 0L + I 0I I · Z M + I 0I + I 0I I · Z C2 0 .

(58)

Fig. 10 Equivalent circuit of the zero (reverse) network sequence at installing the DR on a doublechain line

Improving Methods for One-Sided Determination …

27

  I If the resistance of both circuits is equal Z 0(2) = Z II0(2) = Z 0(2) , which is characteristic of double-chain lines, the currents will be equal in these chains   I I 0(2) = I II0(2) = I 0(2) . With this fact, Eq. (58) will take the following form:   U ϕ0 = I 0 · Z 0 + Z M + 2 · Z C2 0 .

(59)

For the reverse sequence, you can make an equation   U ϕ2 = I 2 · Z 2 + 2 · Z C2 2 .

(60)

From Eqs. (59) and (60), the resistances of the zero and reverse sequences of the C2 system are determined when the central circuit is installed on a double-chain line: Z C2 0

 U Φ0 − Z0 − Z M , I0   U Φ2 1 = · − Z2 . 2 I2

1 = · 2

Z C2 2



(61) (62)

For a direct sequence, the equation compiled according to the corresponding equivalent circuit (Fig. 11):   U ϕ1 − E C2 = I 1 · Z 1 + 2 · Z C2 1 .

(63)

EMF of a direct sequence on the part of the C2 system under the assumption that C2 Z C2 1 ≈ Z2 :   E C2 ≈ U ϕ1 − I 1 · Z 1L + 2 · Z C2 2 .

(64)

Fig. 11 Equivalent circuit of direct network sequence at installing the RD on a double-chain line

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2.2 Calculation of SC Currents from the System of the Opposite End of the Line In the case of a SC on a single-chain line, it can be drew up an equation according to equivalent circuits of the zero and reverse sequences of the network (Fig. 12), from which SC currents of the zero and reverse sequences from the C2 system are determined:   C2 U ϕ0 − I 0 · Z 0l = −I C2 0 · Z 0(L−l) + Z 0 ,

(65)

  C2 U ϕ2 − I 2 · Z 2l = −I C2 2 · Z 2(L−l) + Z 2 .

(66)

From Eqs. (65) and (66) it can be obtained: I C2 0 = I C2 2 =

I 0 · Z 0l − U ϕ0 Z 0(L−l) + Z C2 0 I 2 · Z 2l − U ϕ2 Z 2(L−l) + Z C2 2

,

(67)

,

(68)

where I 0 and I 2 —currents of zero and reverse sequences at the measurement point; U ϕ0 and U ϕ2 —phase voltage of zero and reverse sequences at the measurement point; Z 0l and Z 2l —resistance of zero and reverse sequences from the measurement point to the damage place; Z 0(L−l) and Z 2(L−l) —resistance of the zero and reverse sequences from the damage place to the opposite end of the line.

Fig. 12 Equivalent circuit of the zero (reverse) network sequence in the case of SC on a single-chain line

Improving Methods for One-Sided Determination …

29

The direct sequence current from system C2 (Fig. 13) can be obtained from the equation:   C2 U ϕ1 − I 1l · Z 1l = E C2 − I C2 1 · Z 1(L−l) + Z 1 , I C2 1 =

E C2 − U ϕ1 + I 1l · Z 1l Z 1(L−l) + Z C2 1

(69)

.

(70)

.

(71)

C2 Assuming that Z C2 1 ≈ Z 2 , Eq. (70) takes the form:

I C2 1

=

E C2 − U ϕ1 + I 1l · Z 1l Z 1(L−l) + Z C2 2

In the case of a SC on one of the circuits of a double-chain line (Fig. 14), the calculation of the zero sequence current from the C2 system can be performed based on the equation: U ϕ0 − I 0I · Z 0l − I 0I I · Z Ml = −I 0I I · Z 0(L−l) + I 0I I · Z M(L−l) C2 C2 − I C2 0 · Z 0(L−l) − I 0 · Z 0 .

(72)

The zero sequence current from the C2 system is calculated by the formula I C2 0 =

I 0I · Z 0l + I 0I I · Z M L − I 0I I · Z 0(L−l) − U ϕ0 Z 0(L−l) + Z C2 0

.

(73)

Fig. 13 Equivalent circuit of the direct sequence network in the case of SC on a single-chain line

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Fig. 14 Equivalent circuit of the zero sequence network in the case of SC on a double-chain line

The difference between the equivalent and zero sequence equivalent circuits is the presence of resistance Z M in the zero sequence equivalent circuit, therefore, the expression for the negative sequence current from the C2 system will look like: I C2 2 =

I 2I · Z 2l − I 2I I · Z 2(L−l) − U ϕ2 Z 2(L−l) + Z C2 2

.

(74)

The direct sequence current from the C2 system, according to the equivalent circuit of the direct sequence of the network (Fig. 15), is determined from the equation   C2 C2 II · Z 1(L−l) . U ϕ1 − I 1I · Z 1l = E C2 − I C2 1 · Z1 − I1 + I1

(75)

In this case, the direct sequence current from system C2 is I C2 1 =

E C2 − U ϕ1 + I 1I · Z 1l − I 1I I · Z 1(L−l) Z 1(L−l) + Z C2 1

.

(76)

.

(77)

C2 Assuming that Z C2 1 ≈ Z 2 , Eq. (74) takes the form:

I C2 1 =

E C2 − U ϕ1 + I 1I · Z 1l − I 1I I · Z 1(L−l) Z 1(L−l) + Z C2 2

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31

Fig. 15 Equivalent circuit of the direct sequence network in the case of SC on a double-chain line

2.3 Method for Determining the Distance to the Damage Place Based on the obtained calculated expressions for the LoD by the EMP in cases of one-, two-, and three-phase SC, as well as the calculated expressions for the parameters of the C2 system and its EMP, it seems possible to create a methodology for determining the distance to the damage place of OPL in networks with effectively grounded neutral [33]. For determine the LoD, it is proposed to use a two-stage calculation. The first stage is a preliminary calculation of the distance to the damage place, without taking into account the influence of the C2 system. The second stage is an updated calculation of the distance to the damage place, with taking into account the influence of the C2 system. The peculiarity of the updated calculation is that the refinement of the distance to the place of damage will take place only at the operating mode of the power system at the time of SC is close to the operating mode of the power system at that time point for which the update is being made. Otherwise, a refined calculation may give a result with less accuracy than the previous one. An assessment of the result of the specified calculation should be done by a qualified power engineer who can select from the set of fixed systems of the EMP of system C2 those parameters that most closely correspond to the emergency mode under consideration.

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Two-stage calculation should be done only for damaged lines with double-sided power supply in the presence of C2 system parameters. If information on the parameters of the C2 system is not available, only a preliminary calculation should be made. In single-sided power lines, there is no need to refine the calculation, since the first stage of the calculation gives a result that does not have a methodological error. Let consider the possibility of performing a more accurate calculation using the example of a single-phase EMP with a SC on a double-chain line in the case of SC on a section of a line with a branch. The calculation is performed according to the formula (17) with taking into account (15): Lx = Lab +

−

Re(Uϕ ) − Re( U1 ) Re(I ) − Im(I ) ·

Re(II0 )+Re(III0 )+Re(I0C2 ) Im(II0 )+Im(III0 )+Im(I0C2 )

[Im(Uϕ ) − Im( U1 )] ·

Re(II0 )+Re(I0II )+Re(I0C2 ) Im(II0 )+Im(I0II )+Im(I0C2 )

Re(I ) − Im(I ) ·

Re(II0 )+Re(I0II )+Re(I0C2 ) Im(II0 )+Im(I0II )+Im(I0C2 )

·

·

1 xbd 1 sp 1 xbd 1 sp

,

(78)

where the value I C2 0 of the zero sequence current from the system C2 is determined according to (73). Since I C2 0 depends on the SC, expression (78) is a nonlinear equation. It can be represented as a canonical correlation L x = f (L x ),

(79)

which is usually used to solve equations using mathematical approximation methods. Various methods for the approximate solution of nonlinear equations can be used, which include the methods of simple iteration, Gauss–Seidel, Newton–Raphson and others. The choice of a method for solving the nonlinear Eq. (79) is not a fundamental problem. To implement the refined calculation, it can be used the simplest method for solving nonlinear equations—the simple iteration method. The method ensures the convergence of the refined calculation for single, two, and three-phase SCs.

References 1. De Nooij, M., Koopmans, C., Bijovet, C.: The value of supply security: the costs of power interruptions: economic input for damage reduction and investment in networks. Energy Econ. 29(2), 277–295 (2007). https://doi.org/10.1016/j.eneco.2006.05.022 2. Boonstra, A., de Vries, J.: Analyzing inter-organizational systems from a power and interest perspective. Int. J. Inf. Manag. 25(6), 485–501 (2005). https://doi.org/10.1016/j.ijinfomgt. 2005.08.006 3. Gabriel, S.A., Conejo, A.J., Plazas, M.A., Balakrishnan, S.: Optimal price and quantity determination for retail electric power contracts. IEEE Trans. Power Syst. 21(1), 180–187 (2006). https://doi.org/10.1109/TPWRS.2005.860920

Improving Methods for One-Sided Determination …

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4. Mustafa, M.A., Zhang, N., Kalogridis, G., Fan, Z.: Smart electric vehicle charging: security analysis. In: 2013 IEEE PES Innovative Smart Grid Technologies Conference (ISGT), 24–27 Feb 2013, Washington, DC, USA, pp. 1–6. https://doi.org/10.1109/ISGT.2013.6497830 5. Gupta, A.: Enterprise resource planning: the emerging organizational value systems. Ind. Manag. Data Syst. 100(3), 114–118. https://doi.org/10.1108/02635570010286131 6. Biletskiy, Y., Chikina, V., Yerohin, A., Kaluzhny, D., Senderovich, G.: Decision making support at emergency situations in electric systems. In: Proceedings of the 4th IASTED International Conference on Power and Energy Systems, pp. 199–204 (2014) 7. Biletskiy, Y., Chikina, V., Yerokhin, A., Grib, O., Kaluzhny, D., Senderovich, G.: Methods and models for control of emergency situations in power systems. WSEAS Trans. Syst. 4(8), 1339–1348 (2005) 8. Biletskiy, Y., Chikina, V., Yerokhin, A., Grib, O., Kaluzhny, D., Senderovich, G.: Methods and models for decision-making support at emergency events in power systems. WSEAS Trans. Syst. 4(8), 1349–1354 (2005) 9. Yang, J., Fletcher, J.E., O’Reilly, J.: Short-circuit and ground fault analyses and location in VSC-based DC network cables. IEEE Trans. Ind. Electron. 59(10), 3827–3837 (2012). https:// doi.org/10.1109/TIE.2011.2162712 10. Czapp, S., Borowski, K., Dobrzynski, K., Klucznik, J., Lubosny, Z.: A new method of fault loop resistance measurement in low voltage systems with residual current devices. In: 2015 IEEE Eindhoven PowerTech, 29 June–2 July 2015, Eindhoven, Netherlands, pp. 1–5. https:// doi.org/10.1109/PTC.2015.7232279 11. Aubert, B., Regnier, J., Caux, S., Alejo, D.: Kalman-filter-based indicator for online interturn short circuits detection in permanent-magnet synchronous generators. IEEE Trans. Ind. Electron. 62(3), 1921–1930 (2014). https://doi.org/10.1109/TIE.2014.2348934 12. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Methods and models for information data analysis. In: Diagnostic Systems for Energy Equipments. Studies in Systems, Decision and Control, vol. 281, pp. 23–70 (2020). https://doi.org/10.1007/978-3030-44443-3_2 13. Zaporozhets, A.: Overview of quadrocopters for energy and ecological monitoring. In: Babak, V.P., Isaenko, V.M., Zaporozhets, A.O. (eds.) Systems, Decision and Control in Energy I. Studies in Systems, Decision and Control, pp. 15–36. Springer, Cham (2020). https://doi.org/ 10.1007/978-3-030-48583-2_2 14. Zaporozhets, A., Kovtun, S., Dekusha, O.: System for monitoring the technical state of heating networks based on UAVs. In: Shakhovska, N., Medykovskyy, M.O. (eds.) Advances in Intelligent Systems and Computing IV. CSIT 2019. Advances in Intelligent Systems and Computing, pp. 935–950. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-33695-0_61 15. Babak, S., Babak, V., Zaporozhets, A., Sverdlova, A.: Method of statistical spline functions for solving problems of data approximation and prediction of objects state. In: CEUR Workshop Proceedings, vol. 2353, pp. 810–821 (2019). http://ceurws.org/Vol-2353/paper64.pdf 16. Krasnykov, S.V., Manzhelii, M.I., Lachugin, V.F., Sidoruk, S.V., Dzhangirov, V.A., Boyarkin, I.E.: Experience in using recorders of synchronized current and voltage measurements on 110 kV overhead power lines. Power Technol. Eng. 44, 492–497 (2011). https://doi.org/10.1007/ s10749-011-0213-9 17. Rezinkina, M.M., Rezinkin, O.L., Koliyshko, G.M.: Numerical and experimental investigation of the reliance of high voltage substations’ grounding system in short circuit regimes. In: 1999 Eleventh International Symposium on High Voltage Engineering, 23–27 Aug 1999, London, UK, pp. 411–414. https://doi.org/10.1049/cp:1999067 18. Rezinkina, M.: Numerical modelling of electrical processes in groundings at emergency operation. In: 2011 International Symposium on Lightning Protection, 3–7 Oct 2011, Fortaleza, Brazil, pp. 284–287. https://doi.org/10.1109/SIPDA.2011.6088454 19. Koshkin, I., Kushnir, V., Benyukh, O.: Simulation of 6 (10) kV electrical networks for fault location. In: 2017 International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM), 16–19 May 2017, St. Petersburg, Russia, pp. 1–5. https://doi.org/10.1109/ ICIEAM.2017.8076279

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20. Fedotov, A., Vagapov, G., Basirov, R., Abdulin, L., Grackova, L.: Single-phase ground fault test of overhead power lines in ungrounded power grids of 6–10 kV. In: 2018 IEEE 59th International Scientific Conference on Power and Electrical Engineering of Riga Technical University (RTUCON), 12–13 Nov 2018, Riga, Latvia, pp. 1–5. https://doi.org/10.1109/RTU CON.2018.8659833 21. Kachesov, V.E., Lavrov, VYu., Cherepanov, A.B.: Parametric method of fault location in distribution networks. Power Technol. Eng. 37, 262–268 (2003). https://doi.org/10.1023/A:102639 4109095 22. Joksimovic, G.M., Penman, J.: The detection of inter-turn short circuits in the stator windings of operating motors. IEEE Trans. Ind. Electron. 47(5), 1078–1084 (2000). https://doi.org/10. 1109/41.873216 23. Han, F., Yu, X., Al-Dabbagh, M., Wang, Y.: Locating phase-to-ground short-circuit faults on radial distribution lines. IEEE Trans. Ind. Electron. 54(3), 1581–1590 (2007). https://doi.org/ 10.1109/TIE.2007.894722 24. Vasylets, S., Vasylets, K.: Refinement of the mathematical model of frequency converter cable branch with a singlephase short circuit. East. Eur. J. Enterp. Technol. 4(9(100)), 27–35. https:// doi.org/10.15587/1729-4061.2019.176571 25. Toorani, F., Darabi, A., Salabeigi, I.: ANN-based detection of broken coils of small generator stator with two parallel branches in phase. In: 2008 IEEE 2nd International Power and Energy Conference, 1–3 Dec 2008, Johor Bahru, Malaysia, pp. 1414–1419. https://doi.org/10.1109/ PECON.2008.4762700 26. Jarzhyna, W., Zielinski, D.: The impact of converter’s synchronization during FRT voltage recovery in two-phase short circuits. In: 2015 Selected Problems of Electrical Engineering and Electronics (WZEE), 17–19 Sept 2015, Kielce, Poland, pp. 1–6. https://doi.org/10.1109/ WZEE.2015.7394043 27. German, L.A., Serebryakov, A.S., Osokin, V.L., Subkhanverdiev, K.S.: Equivalent scheme for calculating short-circuit current in an alternating-current electric traction network. Russ. Electr. Eng. 90, 516–521 (2019). https://doi.org/10.3103/S1068371219070058 28. Zhang, J.-H., Chen, X.-Y., Wang, B., Wang, P., Liu, H.-M.: Three-phase short circuit analysis of DFIG-based wind generation and the crowbar biggest resistance setting. In: 2009 International Conference on Sustainable Power Generation and Supply, 6–7 Apr 2009, Nanjing, China, pp. 1–7. https://doi.org/10.1109/SUPERGEN.2009.5348167 29. Alavi, M., Wang, D., Luo, M.: Short-circuit fault diagnosis for three-phase inverters based on voltage-space patterns. IEEE Trans. Ind. Electron. 61(10), 5558–5569 (2014). https://doi.org/ 10.1109/TIE.2013.2297298 30. Kristensen, H., Mrdic, V.: Comparative analysis of three-phase and single-phase dynamic resistance measurement results. In: 22nd International Conference and Exhibition on Electricity Distribution (CIRED 2013), 10–13 June 2013, Stockholm, Sweden, 0473. https://doi.org/10. 1049/cp.2013.0749 31. Kang, N., Liao, Y.: Equivalent PI circuit for zero-sequence double circuit transmission lines. In: 2012 IEEE Power and Energy Society General Meeting, 22–26 July 2012, San Diego, CA, USA, pp. 1–6. https://doi.org/10.1109/PESGM.2012.6344945 32. Abe, R., Taoka, H., McQuilkin, D.: Digital grid: communicative electrical grids of the future. IEEE Trans. Smart Grid 2(2), 399–410. https://doi.org/10.1109/TSG.2011.2132744 33. Taylor, D.: The theory of critical distances. Eng. Fract. Mech. 75(7), 1696–1705 (2008). https:// doi.org/10.1016/j.engfracmech.2007.04.007

Automation of Determining the Location of Damage of Overhead Power Lines Gennadiy A. Senderovich , Artur O. Zaporozhets , Oleg G. Gryb , Ihor T. Karpaliuk , Sergiy V. Shvets , and Inna A. Samoilenko

Abstract Automation of damage location is based on the device and a set of algorithms. The device is built on the basis of a digital oscilloscope based on a computer (alarm recording). It allows to record and store information about the instantaneous values of currents and voltages in three phases of all lines suitable for the substation. Further processing of instantaneous values, performed by the device—analyzer of fixed alarms, gives the desired set of mode parameters at the measurement point. The chapter provides a block diagram of the device. The choice of the moment of the beginning and the duration of the emergency analysis interval is described. The section shows the difference between the developed device and analogs. The main advantage of the developed device is the ability to analyze the modes according to secondary informative parameters obtained after processing the instantaneous values of currents and voltages in the interval of analysis of the emergency mode. In this case, it becomes possible to use the criteria for determining the type of short circuit and the special phase. The block diagrams of algorithms of types of short circuits and a special phase are given. Also given the principles by which the location of a short circuit is determined. Keywords Short circuit · Power lines · Analyzer · Data processing · Algorithms · Block diagrams · Currents · Voltages · Emergency mode parameters

G. A. Senderovich · O. G. Gryb · I. T. Karpaliuk · S. V. Shvets Department of Automation and Cybersecurity, National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine A. O. Zaporozhets (B) Department of Monitoring and Optimization of Thermophysical Processes, Institute of Engineering Thermophysics of NAS of Ukraine, Kyiv, Ukraine e-mail: [email protected] I. A. Samoilenko Department of Management, N.E. Zhukovsky National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 Y. I. Sokol and A. O. Zaporozhets (eds.), Control of Overhead Power Lines with Unmanned Aerial Vehicles (UAVs), Studies in Systems, Decision and Control 359, https://doi.org/10.1007/978-3-030-69752-5_3

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Using a digital oscilloscope (FAS) allows to record and save information about the instantaneous values of currents and voltages in three phases of all lines suitable for the substation. Further processing of instantaneous values by the ANFAS device gives the desired set of mode parameters at the measurement point [1–5]. The high information content of this approach for obtaining EMP and the lack of the need for real-time operation make it possible to use new algorithms to determine the LoD that are inaccessible to microprocessor technology [6, 7]. At the same time, the efficiency of obtaining the result remains one of the most important tasks of the device to determine the LoD. In addition, it is desirable to be able to achieve results without the involvement of highly qualified specialists. In this regard, there is a need for automation of calculations to determine the LoD.

1 Functional Purpose of the BAC BAC is included in the ANFAS fixed alarm analyzer (Fig. 1) and should operate in three modes: automatic; semi-automatic; informational [8–10]. The automatic mode is designed to detect and indicate the type of SC, damaged phases and the distance to the SC location immediately after the line is disconnected and without the participation of operating personnel. The BAC must be launched upon the actuation of the relay protection and must perform the following functions in automatic mode: 1. 2. 3. 4. 5.

selection of the interval analysis emergency mode; determination of the type of SC; determination of a special phase; determination of a damaged line; determination of lines for which the SC is located “behind the back”.

The semi-automatic mode is designed for qualified specialists of power engineers. In this mode, the operator determines the damaged line, the type of SC, the special phase from the waveforms of currents and voltages and sets the interval for analyzing the emergency mode. The task of the BAC in semi-automatic mode is to generate data for calculations to determine the LoD, then they will be automatically used in the corresponding algorithms. Semi-automatic mode allows to increase the accuracy of determining the LoD due to the qualified choice of the beginning and duration of the interval analysis of emergency mode. In addition, in this mode, it can be checked how correctly the BAC works by choosing the calculation interval, determining the SC type and the damaged phase. The information mode is designed to obtain information from this package of mode parameters. The obtained data can be used to verify the operation of algorithms for determining the LoD by, for example, manual counting or for other arbitrary purposes.

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Fig. 1 Flowchart of emergency analyzer

2 Selection of the Start Time and Duration of the Emergency Analysis Interval If the line is disconnected, the FAS saves information about the mode parameters recorded during last 12.8 s. This time covers pre-emergency, emergency and postemergency modes. For determining the LoD, it is necessary to identify the interval of analysis of the emergency mode, characteristic of a stable SC. In this interval, according to the instantaneous values, ANFAS will calculate the effective complex values of the mode parameters averaged over the interval. Information about the emergency mode enters the BAC in the form of a discrete record of instantaneous values of currents and voltages with a discrete interval, i.e. the number of measurements per period, and is set by the mode parameter calculation block [11–15]. The minimum duration of the emergency analysis interval is determined by the required accuracy of converting discrete values of instantaneous periodic values to their effective complex values used in damage location algorithms.

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The end of the analysis interval should occur before the triggering of high-speed protection, the action of which can affect the parameters of the mode that fix. Otherwise, the parameters of various modes may fall into the analysis interval, which will lead to some averaged effective values of currents and voltages, which do not correspond to any real mode, or to a program failure. It is difficult to bind the end of the analysis interval to the moment the protection is triggered, since changes in the mode parameters at the measurement point will not necessarily be qualitative, and quantitative changes may not be sufficient for their use as signs of a mode change. The fixation of the moment of the beginning of the SC can be considered much more reliable. At this point in time, the operation mode of the system changes qualitatively, which is manifested in a strong change in most parameters of the mode and, in particular, the magnitude of the line current. At the beginning of the SC, one cannot ignore the aperiodic component of the SC current. The damping time constant of the aperiodic component of the SCcurrent in 110 kV networks usually does not exceed τa = 0.01 s, in 330 kV networks it can reach τa = 0.05 s. For tuning in time from the aperiodic component of the SC current, it is necessary to shift the beginning of the analysis interval by the time (5–6)τa . In 110 kV networks, this time is comparable to the response time of high-speed protection— high-frequency or the first stage of the remote. It is rather difficult in 110 kV networks to single out a sufficiently long interval for analyzing the parameters of the mode with reproducing its beginning by 0.05–0.06 s from the moment of the beginning of the SC and its end to the moment the protection is triggered. In a 330 kV network, it is generally impossible to fulfill these conditions at reproducing the beginning of the interval 0.25–0.3 s from the beginning of the SC. For ensuring the accuracy of the measurement, it is necessary to exclude the aperiodic component of the SC current [16, 17]. This allows to use the main harmonic component in the analysis of instantaneous values of currents and voltages of bandpass filters. Isolation of the first harmonic is also necessary for detuning from higher harmonics, which can introduce significant errors during SCs through an arc. With ideal filtering of the first harmonic of currents and voltages, there is no need for detuning from the aperiodic components of the SC currents. The algorithm for determining the start time and duration of the analysis interval of the mode parameters is presented in the following form. From the beginning of the recorded alarm file, the instantaneous values of phase currents are scanned. Upon the fact of a sharp change in the instantaneous values of at least one current, the beginning of the SC is recorded. In this case, the determination of the start time of a SC does not require high accuracy. The spread range of the period order T of the fundamental frequency can be considered sufficient. Due to the imperfect nature of the selection of the main harmonic component, the interval of analysis of instantaneous values should be shifted closer to the moment of protection operation. This is necessary for detuning from the not completely filtered aperiodic component. The duration of the analysis interval is taken as the minimum condition for ensuring the measurement accuracy.

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2.1 Determination of the SC Beginning As informative parameters for determining the SC beginning, it is advisable to choose the instantaneous values of currents, since they are expose to more severe changes than voltages [18–20]. For determining the SC beginning, it can be used three elementary equations:  n A0 =

j=n−m

m  n

B0 =

j=n−m

  i B j 

m  n

C0 =

  i A j 

j=n−m

m

  i C j 

K > · 2 K > · 2 K > · 2

n-m j=n−2·m

m n-m j=n−2·m

m n-m j=n−2·m

m

  i A j  ,

(1)

  i B j  ,

(2)

  i C j  ,

(3)

where i A j , i B j , i C j —j-th measurements from the beginning of the scanning of phase currents; m—number of measurements in the averaging interval; n—moving scan interval; K—minimum multiplicity of SC currents. SC can occur in any phase, since the triggering condition is necessary to accept the disjunction of these elementary statements, which corresponds to the equation Y0 = A0 ∨ B0 ∨ C0 = 1.

(4)

Condition (4) means that the SC beginning is fixed upon the fact that K exceeds at least one phase of the average current in the averaging interval from the measurement “n – m” to measurement “n” over the current in the interval from the measurement “n – 2 m” to measurement “n – m”. The averaging interval should be taken equal to the half period. Moreover, the average current value does not depend on the phase shift of the averaging interval. The minimum multiplicity K depends on the value of SC currents. If the launch of the BAC is done upon disconnecting the line, then a high level of selectivity is not necessary. In this case, the value of K can be taken without detuning from the SC outside the zone of protected starting currents, sharp surges of working currents.

2.2 Minimize Emergency Analysis Interval The larger the interval of analysis of instantaneous periodic quantities, the higher the accuracy of the conversion of the values of their discrete measurements into real complex numbers. On the other hand, the duration of the interval is limited by the conditions of the transition process, especially the aperiodic component, and the

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response time of the relay protection. The smaller the measurement interval, the more accurately it can be placed on the time domain of a stable SC. In this regard, it becomes necessary to determine the minimum possible measurement interval. Let consider the dependence of the errors in the calculation of complex quantities on the duration of the analysis interval of discrete measurements of their instantaneous values. The methodological error of measuring the phase of a complex quantity does not depend on the duration of the analysis interval and is reduced to a discreteness error. This issue was resolved during the development of the FAS DR and its solution was confirmed during certification for compliance with the state standards of Ukraine of the ANTES AK-3F electric energy quality indicator based on this registrar. ANTES AK-3F provides a relative error in measuring voltage and current of not more than 0.1%, power—not more than 0.2% [21–23]. The duration of the analysis interval of instantaneous values has a significant effect on the methodological error in calculating the effective values of harmonic signals, especially at times comparable to the period of the fundamental frequency. As the current or effective value of the current (voltage) in electrical engineering, its average value for the period is taken:  I =

1 T



T

i 2 dt.

(5)

0

For harmonic functions: Im I =√ , 2

(6)

where I m —amplitude value of current. Due to the fact that the information in the FAS is recorded in discrete form, the calculation according to formula (5) should be performed using one of the methods for approximate calculation of certain integrals [1, 24, 25]. Such methods include the formulas of rectangles, trapezoids, tangents, Simpson, etc. Formulas differ in different accuracy of calculations with the same number of breakdown sections. As the number of partitions increases, the accuracy of all methods increases. Reducing the error as a whole is not the end in itself of the problem under consideration. We are interested in the dependence of the error on the measurement interval. The nature of this dependence is the same for different formulas. Therefore, it can be used the simplest formula of rectangles to find patterns. Expression (5) for N measurement points written through the formula of rectangles can be represented as follows:  I =

N t · I 2 · sin2 (ω · t · i + ϕ0 ), i=1 m N · t

(7)

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where Δt—measurement step; i—measurement number; ϕ 0 —initial phase of measurement. The angular frequency for discrete measurement is: ω=

2·π , t · M

(8)

where M is the number of measurement points for the period of the harmonic signal. Next, a variable magnitude of the duration of the measurement relative to the period is introduced: k = N / M.

(9)

Taking into account (8) and (9), the formula for determining the effective value takes the form: 

 k·M 2·π 1 (10) · + ϕ0 . I = Im · sin2 i=1 k·M M The relative error in calculating the effective value of the harmonic function taking into account (6) will be determined depending on the multiplicity k from the expression:  δ=

 k·M 2·π 2 2 · + ϕ0 − 1. sin i=1 k·M M

(11)

According to the formula (11), the errors are calculated depending on the multiplicity k. Figure 2 shows the curve of changes in the calculation error for the initial

Fig. 2 Calculating error of the real value depending on the multiplicity (ϕ0 = 0°)

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Fig. 3 Calculating error of the real value depending on the multiplicity (ϕ0 = 30°)

phase ϕ 0 = 0°. The error has a periodic fading character. At k, which is a multiple of a quarter of the period (T /4), the error δ = 0. Calculation of the effective values at other angles (ϕ 0 = 0), for example, at ϕ0 = 30° (Fig. 3), give the same error attenuation but with a multiplicity of half the period (T /2). The duration of the measurement interval from the condition of minimum methodological errors should be a multiple to half period. The minimum duration, measured by the number of measurement points, is N = M/2. δ = f (k, ϕ0 ) = 0 at k = n · M/2,

(12)

where n = 1, 2, 3… The general picture of the change δ = f (k, ϕ 0 ) is presented in a three-dimensional graph (Fig. 4). For excluding the aperiodic component, a duration is needed that will be a multiple of the period T. Let take the duration of the analysis interval of instantaneous values equal to the period of the fundamental frequency T. If the frequency in the power system changes, the duration should accordingly change.

3 Determination of the Type of SC and Special Phase In contrast to the well-known devices for unilateral determination of the LoD such as MIR, MFI, IMF, made on a microprocessor base, the use of digital oscilloscopes, in particular FAS, allows to conduct analysis of modes using secondary informative parameters obtained after processing instantaneous values of currents and voltages in the analysis interval emergency mode. At the same time, it becomes possible to use criteria for determining the SC type and the special phase, which both coincide

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Fig. 4 Dependence of the error in calculating the effective values on the duration and initial phase of the analysis interval δ = f (k, ϕ 0 )

with those used in microprocessor technology and significantly differ from them. So, in the above devices, the SC type and the special phase are determined by the phase relationship of the vectors of currents symmetric components. These criteria in some cases work erroneously. In this regard, in some works, correction of the angular values of the boundaries of the regions of the characteristic arrangement of vectors is proposed, which is also not a guarantee of the correct action in all emergency conditions [26–30]. The following algorithm is proposed to determine the type of fault and the special phase [31]. 1.

Checking for availability SC is carried out for the fact that the current value exceeds the direct current sequence of at least one of the phases of a given setting. ⎧ ⎨ A1: = IA1 > Iset ; B1: = IB1 > Iset ; ⎩ C1: = IC1 > Iset .

(13)

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The current should be taken as the installation, for which it is guaranteed that its component of the direct sequence of the minimum SC current is exceeded. As a first approximation is I set = I nomTA , where I nomTA is the rated current of the measuring current transformers of the OPL. The use of direct sequence currents, in contrast to phase currents, allows one to detuning from longitudinal asymmetry. The condition for confirming the presence of a SC is the disjunction of elementary statements A1, B1, C1: Y1 = A1 ∨ B1 ∨ C1 = 1.

(14)

At Y1 = 0, the absence of SC on the line is fixed. 2.

If condition (14) is satisfied, the presence of an earth fault is determined. With single-phase and two-phase earth faults in networks with an effectively grounded neutral, zero sequence currents significantly exceed the operating currents of normal phases. The presence of an earth fault can be determined from the condition that the current exceeds the zero sequence of the phase current in the undamaged phase. To do this, it is necessary to find out the execution of elementary events: ⎧ ⎨ A2: = I0 > Ia ; B2: = I0 > Ib ; ⎩ C2: = I0 > Ic ,

(15)

the disjunction of which gives a sufficient condition for the presence of SC on the ground: Y2 = A2 ∨ B2 ∨ C2 = 1. 3.

(16)

If condition (16) is fulfilled, it can be determined the type of SC on the ground. To do this, it must be taking into account that with a single-phase circuit (K(1) ), Eqs. (15) will be performed in two phases, and with a two-phase circuit (K(1,1) ), in one phase. It can be distinguished K(1) and K(1,1) by considering the 3 conjunction equations: ⎧ ⎨ A3: = B2 ∧ C2; B3: = A2 ∧ C2; ⎩ C3: = A2 ∧ B2.

(17)

The condition for the presence of a single-phase is the disjunction of events A3, B3, C3:

Automation of Determining the Location of Damage of Overhead Power Lines

Y3 = A3 ∨ B3 ∨ C3 = 1 → K(1) .

45

(18)

The full condition should take into account that at K(1) , Eq. (15) are satisfied in two phases, and in the third they are not satisfied:       A3 ∧ A2 ∨ B3 ∧ B2 ∨ C3 ∧ C2 → K(1) .

(19)

The determination of the damaged phase in single-phase SC is already inherent in formulas (18), (19). Necessary  sufficient   conditions   for the presence of  and (1) (1) (1) a single-phase SC in phases A K A , B K B , C KC are: ⎧    ⎪ ⎪ A2 ∧ B2 ∧ C2 ∧ B2 ∧ A2 ∧ C2 ∧ ⎪ ⎨    B2 ∧ A2 ∧ C2 ∧ A2 ∧ B2 ∧ C2 ∧ ⎪  ⎪    ⎪ ⎩ C2 ∧ A2 ∧ B2 ∧ A2 ∧ B2 ∧ C2 ∧

  C2 ∧ A2 ∧ B2 → K A(1) ;   C2 ∧ A2 ∧ B2 → K B(1) ;   B2 ∧ A2 ∧ C2 → K C(1) . (20)

Expressions (19) and (20) carry redundant information because the events A2, B2, C2 are not independent. At condition (16) is fulfilled, Y2 = 1, to determine the damaged phase, it suffices to know how conditions (17) are satisfied: ⎧ (1) ⎨ B2 ∧ C2 → K A ; A2 ∧ C2 → K B(1) ; ⎩ A2 ∧ B2 → K C(1) .

(21)

⎧ (1) ⎨ (I0 > I B ) ∧ (I0 > IC ) → K A ; (1) (I > I A ) ∧ (I0 > IC ) → K B ; ⎩ 0 (I0 > I A ) ∧ (I0 > I B ) → K C(1) .

(22)

With regard to (15):

If, in the presence of a ground fault (Y2 = 1), Eq. (18) is not satisfied, then a two-phase SC on the ground takes place. Condition for this damage: Y3 = A3 ∨ B3 ∨ C3 = 0 → K(1,1)

(23)

In the special phase for K(1,1) , one of the Eq. (15) will be satisfied. The definition of a special phase can be described by the following expression: ⎧ (1,1) ⎨ I0 > I A → A (K BC ); (1,1) ); I > I B → B (K AC ⎩ 0 (1,1) ). I0 > IC → C (K AB

(24)

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4.

In the absence of an earth SC (Y2 = 0), it can be determined the type of interphase SC and a special phase, if there is a two-phase SC (K(2) ). Checking for asymmetry can be performed with the condition: Y4: = 6 · I2 > I1 .

(25)

The fulfillment of condition (25) is a sign of asymmetry. Two-phase fault occurs at: Y4 = 1 → K (2) .

(26)

With a two-phase SC in the undamaged phase, the current vectors of the forward and reverse sequences have a mutual angle greater than 180°, in damaged phases this angle is less than 180°. This gives an indication of a special phase: ⎧ (2) ⎨ I2 > I A → A (K BC ); (2) I > I B → B (K AC ); ⎩ 2 (2) ). I2 > IC → C (K AB

(27)

A three-phase SC will be observed if condition (26) is not satisfied: Y4 = 0 → K (3),

(28)

or none of the inequalities (27) is not satisfied: Y5 = (I2 > I A ) V (I2 > I B ) V (I2 > IC ) = 0 → K (3) .

(29)

The block diagram of the algorithm for determining the type of SC and the special phase is shown in Figs. 5, 6 and 7.

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Fig. 5 Block diagram of the algorithm for determining the type of SC and the special phase (part I)

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Fig. 6 Block diagram of the algorithm for determining the type of SC and the special phase (part II)

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Fig. 7 Block diagram of the algorithm for determining the type of SC and the special phase (part III)

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4 Determination of Location of SC For each line, the presence of a fault on it is determined by criterion (14). This estimate is non-selective. Talking about the accuracy of determination is possible only taking into account the disconnection of this line by the protection [32, 33]. In the event of an accident on one of the lines suitable for substations, it will be disabled by relay protection and criterion (14) will be fulfilled for it. For the relay protection of lines that fit the same substation buses as the damaged one, a SC fault will be observed “behind the back”. Modes in which damage is detected “behind the back” of the line relay protection are used to obtain initial data at determining system parameters from the other end of the line. Using the fact of disconnecting lines, or other indirect signs, it is possible to enter into the calculation of parameters information that allows to determine lines for which there is a SC “behind the back”. But such information for each line can also be obtained directly from the parameters of its mode by analysis of EMP. In solving this problem, it is advisable to use direct sequences of symmetrical components, because they are in all types of SCs. For resistive-inductive circuits, which are characteristic of power supply networks with an effectively grounded neutral, the direct-sequence current lags behind the voltage (Fig. 8a). In 110 kV networks—ϕ1 ≈ 70°. In the case of a SC “behind the back”, the nature of the network is practically the same, but the current has the opposite direction (Fig. 8b). In this case, the angle changes by 180°. In 110 kV networks, ϕ1 ≈ 70° + 180°. From here the criterion is appeared. If in the special phase ϕ1 < π/2, then the SC is located on the line side. For ϕ1 > π/2, the SC is “behind the back”.

Fig. 8 Determination of the location of the short circuit: a from the side of the line; b “behind the back”

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References 1. Babak, V., Eremenko, V., Zaporozhets, A.: Research of diagnostics parameters of composite materials using Johnson distribution. Int. J. Comput. 18(4), 483–494 (2019) 2. Eremenko, V., Zaporozhets, A., Isaenko, V., Babikova, K.: Application of wavelet transform for determining diagnostic signs. In: CEUR Workshop Proceedings, vol. 2387, pp. 202–214. https://ceur-ws.org/Vol-2387/20190202.pdf 3. Zaporozhets, A., Babak, V., Sverdlova, A., Isaienko, V., Babikova, K.: Development of a system for diagnosing heat power equipment based on IEEE 802.11s. In: Zaporozhets, A., Artemchuk, V. (eds.) Systems, Decision and Control in Energy II. Studies in Systems, Decision and Control, pp. 141–151. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-69189-9_8 4. Kovalev, G.F., Lebedeva, L.M.: Programs for analyzing the reliability of EPS, conditions, and basic provisions of their application to the design practice. In: Reliability of Power Systems. Power Systems. Springer, Cham, pp. 67–111 (2019). https://doi.org/10.1007/978-3-030-187 36-1_3 5. Parandehgheibi, M., Turitsyn, K., Modiano, E.: Modeling the impact of communication loss on the power grid under emergency control. In: 2015 IEEE International Conference on Smart Grid Communications (SmartGridComm), 2–5 Nov. 2015, Miami, FL, USA, pp. 356–361. https://doi.org/10.1109/SmartGridComm.2015.7436326 6. Negnevitsky, M., Voropai, N., Kurbatsky, V., Tomin, N., Panasetsky, D.: Development of an intelligent system for preventing large-scale emergencies in power systems. In: 2013 IEEE Power & Energy Society General Meeting, 21–25 July 2013, Vancouver, BC, Canada, pp. 1–5. https://doi.org/10.1109/PESMG.2013.6672099 7. Popov, M.G., Petrushin, D.E., Efimov, N.S.: Increasing the emergency control systems efficiency in Kola’s and Karelia’s power systems. In: 2019 IEEE Conference of Russian Young Researchers in Electrical and Electronic Engineering (EIConRus), 28–31 Jan, 2019, Saint Petersburg and Moscow, Russia, pp. 1040–1043. https://doi.org/10.1109/EIConRus.2019.865 7093 8. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Simulation and software for diagnostic systems. In: Diagnostic Systems for Energy Equipments. Studies in Systems, Decision and Control, vol. 281, pp. 71–90. Springer, Cham (2020). https://doi.org/ 10.1007/978-3-030-44443-3_3 9. Zuev, A., Gryb, O., Shvets, S., Makarov, V.: Evaluating and ensuring the cybersecurity of power line remote monitoring systems. In: 2018 IEEE 3rd International Conference on Intelligent Energy and Power Systems (IEPS), 10–14 Sept. 2018, Kharkiv, Ukraine, pp. 271–274. https:// doi.org/10.1109/IEPS.2018.8559572 10. Kuchanskyy, V., Malakhatka, D., Blinov, I.: Application of reactive power compensation devices for increasing efficiency of bulk electrical power systems. In: 2020 IEEE 7th International Conference on Energy Smart Systems (ESS), pp. 83–86 (2020). https://doi.org/10.1109/ ESS50319.2020.9160072 11. Rezinkin, O., Rezinkina, M., Nosenko, M.: Usage of the spiral pulsar in the high voltagecurrent generator system. In: IEEE Conference Record—Abstracts. PPPS-2001 Pulsed Power Plasma Science 2001. 28th IEEE International Conference on Plasma Science and 13th IEEE International Pulsed Power Conference (Cat. No.01CH37), 17–22 June 2001, Las Vegas, NV, USA, pp. 439. https://doi.org/10.1109/PPPS.2001.961196 12. Rezinkin, O.L., Rezinkina, M.M., Gryb, O.G.: Synthesis of ferroceramics for electromagnetic shock waves generators by vacuum aerosol deposition method. Funct. Mater. 23(3), 484–489 (2016). https://doi.org/10.15407/fm23.03.484 13. Rezinkin, O.L., Rezinkina, M.M., Gryb, O.G., Revutsky, V.I.: Cold pressing of ferroelectricferromagnetic layered composites for nonlinear forming lines of high-voltage impulse generators. Funct. Mater. 24(1), 168–174 (2017). https://doi.org/10.15407/fm24.01.168 14. Schon, K.: High alternating voltages and currents. In: High Voltage Measurement Techniques. Power Systems. Springer, Cham, pp. 7–42 (2019). https://doi.org/10.1007/978-3-03021770-9_2

52

G. A. Senderovich et al.

15. Schon, K.: High direct voltages and currents. In: High Voltage Measurement Techniques. Power Systems. Springer, Cham, pp. 43–77 (2019). https://doi.org/10.1007/978-3-030-21770-9_3 16. Boutsika, T.N., Papathanassiou, S.A.: Short-circuit calculations in networks with distributed generation. Electric Power Syst. Res. 78(7), 1181–1191 (2008). https://doi.org/10.1016/j.epsr. 2007.10.003 17. Ye, L., Campbell, A.M.: Case study of HTS resistive superconducting fault current limiter in electrical distribution systems. Electric Power Syst. Res. 77(5–6), 534–539. https://doi.org/10. 1016/j.epsr.2006.05.007 18. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Methods and models for information data analysis. In: Diagnostic Systems for Energy Equipments. Stud. Syst. Decis. Control 281, 23–70 (2020). https://doi.org/10.1007/978-3-030-44443-3_2 19. He, Z., Zhong, C., Huang, X., Wong, W.-Y., Wu, H., Chen, H., Chen, L., Su, S., Cao, Y.: Simultaneous enhancement of open-circuit voltage, short-circuit current density, and fill factor in polymer solar cells. Adv. Mater. 23(40), 4636–4643. https://doi.org/10.1002/adma.201103006 20. Lu, B., Sharma, S.K.: A literature review of IGBT fault diagnostic and protection methods for power inverters. IEEE Trans. Ind. Appl. 45(5), 1770–1777 (2009). https://doi.org/10.1109/ TIA.2009.2027535 21. Dedinec, A., Filiposka, S., Dedinec, A., Kocarev, L.: Deep belief network based electricity load forecasting: an analysis of Macedonian case. Energy 115(3), 1688–1700 (2016). https:// doi.org/10.1016/j.energy.2016.07.090 22. Catrinu, M.D., Nordgard, D.E.: Integrating risk analysis and multi-criteria decision support under uncertainty in electricity distribution system asset management. Reliab. Eng. Syst. Saf. 96(6), 663–670 (2011). https://doi.org/10.1016/j.ress.2010.12.028 23. Amjady, N., Keynia, F.: Electricity market price spike analysis by a hybrid data model and feature selection technique. Electric Power Syst. Res. 80(3), 318–327. https://doi.org/10.1016/ j.epsr.2009.09.015 24. Babak, S., Babak, V., Zaporozhets, A., Sverdlova, A.: Method of statistical spline functions for solving problems of data approximation and prediction of objects state. In: CEUR Workshop Proceedings, vol. 2353, pp. 810–821 (2019). https://ceur-ws.org/Vol-2353/paper64.pdf 25. Zaporozhets A., Eremenko V., Serhiienko R., Ivanov S.: Methods and hardware for diagnosing thermal power equipment based on smart grid technology. In: Shakhovska, N., Medykovskyy, M. (eds) Advances in Intelligent Systems and Computing III. CSIT 2018. Advances in Intelligent Systems and Computing, vol 871. Springer, Cham, pp. 476–489 (2019). https://doi.org/ 10.1007/978-3-030-01069-0_34 26. Zaporozhets, A.A., Eremenko, V.S., Serhiienko, R.V., Ivanov, S.A.: Development of an intelligent system for diagnosing the technical condition of the heat power equipment. In: 2018 IEEE 13th International Scientific and Technical Conference on Computer Sciences and Information Technologies (CSIT), 11–14 Sept. 2018, Lviv, Ukraine, pp. 48–51. https://doi.org/10.1109/ STC-CSIT.2018.8526742 27. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Technical provision of diagnostic systems. In: Diagnostic Systems for Energy Equipments. Studies in Systems, Decision and Control, vol. 281, pp. 91–133 (2020). https://doi.org/10.1007/978-3030-44443-3_4 28. Krejcar, O., Frischer, R.: Real time voltage and current phase shift analyzer for power saving applications. Sensors 12(8), 11391–11405 (2012). https://doi.org/10.3390/s120811391 29. Kikkert, C.J., Zhu, S.: Resistive shunt on-line impedance analyzer. In: 2016 International Symposium on Power Line Communications and its Applications (ISPLC), 20–23 March 2016, Bottrop, Germany, pp. 150–155. https://doi.org/10.1109/ISPLC.2016.7476269 30. Lu, Y., Liu, W.: Spectrum analyzer based measurement and detection of MW/SW broadcast radios on power lines for cognitive PLC. In: 2013 IEEE 17th International Symposium on Power Line Communications and Its Applications, 24–27 March 2013, Johannesburg, South Africa, pp. 103–108. https://doi.org/10.1109/ISPLC.2013.6525833 31. Rosen, K.H., Krithivasan, K.: Discrete Mathematics and its Applications: With Combinatorics and Graph Theory. Tata McGraw-Hill Education (2012)

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32. Blinov, I., Zaitsev, I.O., Kuchanskyy, V.V.: Problems, methods and means of monitoring power losses in overhead transmission lines. In: Babak, V.P., Isaenko, V.M., Zaporozhets, A.O. (eds.) Systems, Decision and Control in Energy I. Studies in Systems, Decision and Control, pp. 123– 136. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-48583-2_8 33. Yablokov, A., Filatova, G., Timofeev, A.: Using of non-traditional current and voltage sensors for the fault location. In: MATEC Web of Conferences, vol. 141, p. 01058 (2017). https://doi. org/10.1051/matecconf/201714101058

Experimental Studies of the Method for Determining Location of Damage of Overhead Power Lines in the Operation Mode Gennadiy A. Senderovich , Artur O. Zaporozhets , Oleg G. Gryb , Ihor T. Karpaliuk , Sergiy V. Shvets , and Inna A. Samoilenko Abstract The chapter presents the results that have received experimental confirmation of the operability of the method of one-sided determination of the location of damage. The analysis was carried out on data under operating conditions in 110 kV networks. Damage locations were determined by the emergency mode parameters of the power transmission lines of a number of emergency shutdowns of the lines recorded during the operation of the “TsPRS” and “Rekon-06BS” software and hardware systems, according to the method of one-sided determination of the location of the damage. The chapter shows a block diagram of the hardware-software complex “TsPRS”, describes the principles of the complex. It shows a block diagram of a unit for determining the location of the damage, which is designed to calculate the distance to the place of damage, and a unit for determining the parameters of the system, which is designed to calculate the parameters of the system. A step-by-step analysis of the operation of the device is given in the analysis of emergency blackouts of power lines in 110 kV networks. Keywords Software · Hardware · Block diagram · Power transmission lines · Emergency blackouts · Emergency parameters · Location of damage · Short circuit · UAVs · 110 kV networks

G. A. Senderovich · O. G. Gryb · I. T. Karpaliuk · S. V. Shvets Department of Automation and Cybersecurity, National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine A. O. Zaporozhets (B) Department of Monitoring and Optimization of Thermophysical Processes, Institute of Engineering Thermophysics of NAS of Ukraine, Kyiv, Ukraine e-mail: [email protected] I. A. Samoilenko Department of Management, N.E. Zhukovsky National Aerospace University “Kharkiv Aviation Institute”, Kharkiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 Y. I. Sokol and A. O. Zaporozhets (eds.), Control of Overhead Power Lines with Unmanned Aerial Vehicles (UAVs), Studies in Systems, Decision and Control 359, https://doi.org/10.1007/978-3-030-69752-5_4

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Fig. 1 Photos of a 330 kV OPL with UAV: a general view; b OPL’s tower; c power line insulators

Fig. 2 Photos of the electrical equipment of a 110 kV substation in the infrared spectrum: a 110 kV high-voltage switches, b disconnector

The development of the one-sided LoD presented above was confirmed by analyzing the data obtained under operating modes. The ability to analyze emergency shutdowns [1–5] of 110 kV overhead lines was provided by the authors at Kharkivoblenergo. This allowed to put into the basis of verification of theoretical developments the data on the industrial operation of electric networks with an effectively grounded neutral during diagnosing the elements of the electric network in the optical range using UAVs (Fig. 1). A separate type of diagnosis of electrical equipment is the researches in the infrared spectrum. In the works [6–10], data from the inspection of power lines in the infrared range are presented (Figs. 2 and 3). The chapter presents the results of the analysis of a one-sided LoD by EMP of OPL in 110 kV networks of a number of emergency shutdowns of lines detected during the operation of the TsPRS and Rekon-06BS hardware-software complexes. The damage

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Fig. 3 Photo in the infrared spectrum of the elements of the 110 kV power transformer

Fig. 4 ANTES AR-3F power network analyzer

location of the examined lines is precisely established during their inspection. The analysis of emergency shutdowns of the lines was carried out using the ANFAS emergency process software, implemented in the ANTES AR-3F electric network mode analyzer (Fig. 4). For compatibility of “TsPRS” and “Rekon-06BS” data, the general format for the exchange of transient data in energy systems (COMTRADE) of the IEEE standard was used [11, 12].

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1 Hardware-Software Complex “TsPRS” The hardware-and-software complex “TsPRS” is intended for registration of emergency processes in electric power systems, creation of emergency process recording files and transfer of these files to a remote user for their subsequent analysis, processing and, if necessary, documentation. “TsPRS” consists of an autonomous digital alarm recorder “FAS” and the software “ANFAS” (Fig. 5). The registration of the parameters of the electric line is carried out using the DR “FAS” located at the electrical substation. Constantly monitoring the parameters of the controlled object modes (voltage of bus sections and currents of outgoing lines from the substation) is carried out [13, 14]. At the event of an emergency mode, the FAS generates, records and saves a file of the emergency process, which includes the pre-emergency, emergency and post-emergency modes. The device is implemented on the basis of a personal computer with a processor of class 80,386. The computer is working on Linux operating system. The device registers and stores in its memory the instantaneous values of currents and voltages of the electric line. The sampling frequency of the device is 5 kHz, which provides sufficient accuracy during processing recorded signals. The parameters of both the normal mode of operation of the electric line and the parameters of the emergency mode are recorded. The

Fig. 5 Block diagram of the hardware and software complex TsPRS

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Table 1 General characteristics of hardware and software “ANFAS” Data processing location

Electrical substation

Control room

CPU

Not less i386

Pentium I and higher

RAM

4–16 MB

Not less 16 MB

HDD

800 MB

Not less 1 MB for software

“FAS” device board

Need

No need

Operating system

Linux

Windows 9x/ME/NT/2000/XP

Software

Archive tar

“AnFAS”

Hardware

Software

recording duration of the entire file is 12.8 s, of which the first 0.4 s is the preemergency mode. Data is written to a file in a special format, then archived to reduce the size for transmission over the network. Further signal processing can be performed both at the substation itself and at a remote control center. In this case, the recorded data must be transferred to the control room. There are no special technical requirements for data transfer because modem communication is used. Determination of operation’s modes parameters of electric lines and related applied problems are solved using special software. Some of these tasks are solved using the ANFAS software product. The ANFAS program runs on personal computers which work on the Windows operating system family. The system requirements of the program are not large: free disk space—1 Mb; RAM capacity—16 Mb; central processor—class 80,486 and higher. For comfortable work with the program, it needs a color monitor with a resolution of at least 800 × 600 and a mouse. Depending on the place of processing the data recorded in the FAS, the configuration of the computer and software required for the implementation of ANFAS is slightly different. The structure and necessary requirements for the operation of the system are given in Table 1. The ANFAS program is designed to view and analyze the parameters of the electric line recorded using the “FAS” instrument. In addition to the file format recorded using “FAS”, the program can view and analyze files recorded in the COMTRADE format (a common format for transient data exchange in power systems. IEEE C37.111–1191 standard), which expands the possibilities of its use. Viewing the data of emergency and normal modes occurs in the form of graphs (oscillograms), which represent the instantaneous values of currents and voltages of the electric line. For ease of viewing, the functions of values and time scaling, selective display of graphs, manual and automatic scrolling of images, the use of labels, saving screens, exporting screens to graphic format files, selecting and editing the color scheme of graph representations are available. Data processing tools include procedures for determining the parameters of an electric line based on available data. These parameters include the frequency of the

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current values of the currents and voltages of the channels, the phase relationships of the channels [15–19]. Such operations as: building vector diagrams of signals, decomposing signals into harmonics, determining the symmetrical components of the signals, etc. are also available. An important feature of the program is the built-in error correction capabilities of recorded data. One of the main tasks that can be solved with the help of the program is determining the LoD in the event of a SC on the line. In this regard, there are special functions for automatically searching for the place of an accident, determining the type of SC and calculating its parameters with the subsequent determination of the place of damage. The same features are available in manual mode by the operator. To accomplish the task, “ANFAS” contains two main parts: a calculating device of emergency mode parameters (CDEMP) and a device for determining the location of damage (DLoD) (Fig. 5). CDEMP is intended for processing instantaneous values of currents and voltages recorded by the DR, as well as for the preparation and formation of initial data for the DLoD. The solution to the problem of determining the location of damage to the electric line is implemented in the DLoD. The algorithm of the device incorporates the LoD method. During determining the place of damage, the parameters of the electric network mode are used, obtained by calculation in the CDEMP, as well as the parameters of the electric network saved in the configuration file. DLoD is designed to calculate the distance to the place of damage. The following basic blocks are included in the composition of the DLoD (Fig. 5): • damage location block (DLB) is designed to calculate the distance to the place of damage; • block of determining the parameters of the C2 system (BDPCS) is intended for calculating the parameters of the C2 system in cases where a SC occurred “behind the back”. Other DLoD blocks have auxiliary character; their purpose is shown in Fig. 5. DLB includes three independent units designed for the LoD in cases of one-, twoor three-phase SC [20, 21]. An enlarged block diagram of the DLB is shown in Fig. 6. Initial information for DLB are: SC type; EMPs; parameters of the damaged line. Depending on the SC type, a calculation algorithm is selected, which corresponds to the expression for determining the distance with this damage. The use of BDPCS allows to calculate the parameters of the system connected to the opposite end of the line. In this case, the mode parameters recorded on the line under consideration during SC on other lines suitable for the common bus, or more obvious damage, are used. Determination of EMF and resistances of all sequences from the results of one measurement is possible in the case when SC connected to the ground [22]. Considering that the largest number of SCs connected with earth is in single-phase SCs, it is advisable to base the algorithm for determining the parameters of the C2 system on this type of SC. In the block diagram of the algorithm for determining the parameters of the C2 system (Fig. 7), the SC position “behind the back” is determined by the phase displacement between the current and voltage vectors of the direct sequence ϕ1 , and corresponds to the condition ϕ1 > π.

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Fig. 6 DLB block diagram

The block diagram is presented for the case of a double-stranded line, and assumes the presence of magnetic bonds in the zero sequence. If measurements are taken on a single-circuit line, appropriate simplifications must be made. The following quantities act as the initial data of the considered algorithm: phase voltages and currents of the forward, reverse, and zero line sequences; line configuration; matrix of line parameters. After calculating the parameters of the C2 system, they are stored in the database with the date and time of the state of the system.

2 Analysis of Emergency Blackouts of Power Lines in 110 kV Networks The analysis of emergency shutdowns [23–26] recorded during the operation of the hardware and software systems “TsPRS” installed at the substation “SiM” and substation “I”, as well as “Rekon-06BS”—at the substation “K”. The complexes control the voltages of all bus sections and the currents of all substation lines on which they are installed. The names of the substations indicated as abbreviations due to the fact that service information was used.

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Fig. 7 BDPCS block diagram

2.1 Damage to the “L-SiM” Overhead Line According to the record of the operational journal 21/03/00 at 14:39 at the line “L-SiM” there was a single-phase SC. Phase A was damaged. The reason for disconnecting the overhead line is a SC of the L-SiM line with a lightning protection cable at the intersection of its 330 kV overhead line. The distance to the place of damage is 11.31 km from substation “SiM”. Description of the damaged line. The “L-SiM” line connects the “SiM” substation and the “L” substation, and is double-circuit with a branch to the “GPZ 8” substation. The “L-SiM” line has intersections with a 330 kV overhead line. The total length of the line from “SiM” substation to “L” substation is 12.563 km; the distance from “SiM” substation to the branch is 9.784 km; the length of the branch is 2 km; the distance from “SiM” substation to the intersection with a 330 kV OPL is 11.31 km [27, 28]. Figure 8 shows the line equivalent circuit. In pre-emergency mode, both circuits of the damaged “L-SiM” line are switched on and operate in parallel. The transformer neutrals at the “GPZ 8” substation are

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Fig. 8 “L-SiM” line equivalent circuit

open. The oscillogram of the emergency shutdown [29, 30] of the line is shown in Fig. 9. In this record of the emergency shutdown of the line, three characteristic stages can be distinguished. Stage 1—single-phase SC on the “L-SiM” line № 1 (Fig. 10a). The first stage of the accident ends with the disconnection of the “L-SiM” line № 1 from the other end—from the substation “L”. A characteristic feature of the 1st stage of the accident is the feeding of the SC on both sides of the line—from the “SiM” substation and from the “L” substation. Stage 2—continuation of a single-phase SC on the “L-SiM” line № 1 with switch off from the “L” substation side (Fig. 10b). The second stage ends with the disconnection of the “L-SiM” line № 1. A characteristic feature of the 2nd stage of the accident is the feeding of a SC on one side of the line—from the “L-SiM” line.

Fig. 9 Complete oscillogram of emergency shutdown of the “L-SiM” overhead line

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Fig. 10 Design scheme of the line: a the first stage; b the second stage; c the third stage

Stage 3—single-phase SC on the “L-SiM” line № 2 (Fig. 10c). The third stage ends with the disconnection of the “L-SiM” line № 2. A characteristic feature of the 3rd stage of the accident is the feeding of a SC at the “L-SiM” line № 2 from two sides of the line with the parallel “L-SiM” line № 1 line disconnected. Let determine the LoD on the “L-SiM” line using the automatic, semi-automatic and informational ANFAS operating modes. Automatic operation mode. Automatically detected: damaged line—“L-SiM” № 1, type of SC—single-phase, damaged phase A and calculating interval EMP, which shown in Fig. 11. As a result of this mode of operation, two distances to the place of damage were obtained. The first is the preliminary distance to the place. The second is the specified distance to the place of damage according to the state of the C2 system on 21/03/00 14:39. The preliminary distance was 11.06 km, the specified distance was 11.18 km, while the relative linear deviation of the preliminary distance from the real value was 2.2%, and the deviation of the specified value was 1.1%.

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Fig. 11 Calculated interval of EMP in automatic operation at analyzing the emergency shutdown record of the “L-SiM” overhead line

Semi-automatic operation mode. In the semi-automatic mode of operation, two stages of the accident were considered (Fig. 12), the first and second, for which the calculated EMP intervals were set manually. At choosing the calculating intervals of the EMP, the optimal intervals of the established modes of SC were determined. 1st stage of the accident. The duration of the calculation interval of the EMP is 0.05 s, the deviation from the initial moment of the SC is 0.041 (Fig. 12). The preliminary distance to the place of damage was 11.19 km (deviation of 1%), the specified distance was—11.3 km, on 21/03/00, 14:39 (deviation of less than 1%). 2nd stage of the accident. Considering that the feature of the second stage of the accident is the feeding of the SC on one side of the line, in the semi-automatic mode of operation, only the preliminary distance to the place of damage was calculated. The duration of the calculation interval of the EMP is 0.05 s, the deviation from

Fig. 12 Calculating intervals of EMP at semi-automatic operation during analysis of emergency shutdown recording of “L-SiM” overhead lines

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the beginning of the SC is 0.125 s (Fig. 12). The distance to the damage point was 11.2 km (deviation—1%). Information operation mode. In the information mode of operation, as well as in the semi-automatic mode, the first and second stages of emergency shutdown of the line were considered [31]. The result of the information mode is shown in Table 2 for the first and second stages of the accident. The calculated EMP intervals are selected similarly in the semi-automatic mode of operation. Complex EMP values are given relative to the voltage vector of the damaged phase directed along the real axis. Based on the obtained EMP values, it will be calculated the preliminary and specified distance to LoD according to the first stage of the emergency shutdown of the line. Preliminary calculation. Suppose that the LoD is on the site from the “SiM” substation to the branch at the “GPZ-8” substation. Because in the general case it is not known where the point of the SC is located (before or after the place of branching), then at first it can be accepted that the SC is before the branch. Then, for determining the preliminary distance to the place of damage, it can be used the calculated expression (17, Chap. 2), in which U1 = 0, and the value of ctgϕk is determined by the expression (16, Chap. 2). The zero sequence current and voltage vectors have the following meanings: • zero sequence current of the damaged line I 0I = 0.937 ∠ −62.95 kA; • zero sequence voltage of the damaged line U 0I = 5.68 ∠ −162.5 kV; • zero sequence current of the parallel line I II0 = 0.337 ∠ −65.5 kA. Communication coefficients are determined according to the expression (9, Chap. 2) according to the line parameters provided in Fig. 8: KR = (2 · 1.55 + 3.34)/3.51 = 1.837; Kx = 2 + 12.61/3.51 = 5.593; KMR = 1.09/3.51 = 0.311; KMX = 8.1/3.51 = 2,307; KR1 = 1.55/3.51 = 0.441. Specific reactance: Xsp = 3.51/9.784 = 0.3587 Om/km. The components of expression (17, Chap. 2):   Re ΔI 1ph = 1.62 − 3 · 0.426 = 0.336;   Im ΔI Iph = −3.08 − 3 · (−0.834) = −0.579;   Re I x = 0.426 · 1.834 − (−0.835) · 5.593 + 0.125 · 0.31 − (−0.312) · 2.307 



+ 0.336 · 0.442 − (−0.579) = 6.94;

Im I x = 0.426 · 5.593 + (−0.835) · 1.834 + 0.125 · 2.31 + (−0.312) · 0.31 + 0.336 + (−0.579) · 0.441 = 1.125; ctg φk = (0.4266 + 0.125 + 0)/(−0.835 − 0.313 − 0) = −0.481.

0

46.62

0

Effective value (kV, kA)

Phase, o

II stage emergency mode

0.49

Phase, o

58.44

244.3

61.84

253.1

59.92

120.5

61.62

120.4

3.48

0 0

6.7

110.6

0.37

−66.1

297.6

Ib 0.31

144.6

0.018

131.6

Ic

Ia

Uc

Ua

Ub

Phase currents of damaged line

Phase voltages

Effective value (kV, kA)

I stage emergency mode

Mode parameters

Table 2 Result of the emergency shutdown information mode of the “L-SiM” overhead line

97.4

0.4

290.8

1.72

Ia

86.49

0.532

99.19

0.399

Ib

Phase currents of parallel line

0.33

116.9

0.567

121.6

Ic

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Preliminary distance to the place of damage: LX =

1 30.49 − 0 · (−0.481) · = 11.363 km 6.94 − 1.125 · (−0.481) 0.3587

The obtained value exceeds the distance to the branch (11.363 km > 9.784 km), therefore, the place of damage is outside the considered section of the line. Therefore, let consider the next variant. Suppose that the place of damage is located on a section of the line from the “GPZ-8” substitution to the “L” substitution. Due to the fact that the neutrals of the transformers at the “GPZ-8” substation are open, it can be taken the zero sequence current before the branch to be equal to the zero sequence Ibc current after the branch (I Iab 0 = I 0 ). Neglecting the branch load currents, phase the current before the branch, as for the zero sequence currents, can be taken equal to the Ibc phase current after the branch (I Iab ϕ = I ϕ ). The calculation is performed according to the formula (17, Chap. 2), in which it can be determined the value of ctgϕk by the expression (15, Chap. 2). Communication coefficients: KR = (2 · 0.49 + 1.065)/1.12 = 1.825; KX = 2 + 4.02/1.12 = 5.589; KMR = 0.57/1.12 = 0.51; KMX = 2.56/1.12 = 2.586; KR1 = 0.49/1.12 = 0.438.

Specific reactance: Xsp = 3.51/9.784 = 0.3587 Om/km. The components of expression (17, Chap. 2):     = Re ΔI lbc = 0.336; Re ΔI lab ϕ ϕ     = Im ΔI Ibc = −0.579; Im ΔI lab ϕ ϕ rΣ = 2 · 1.55 + 3.34 = 6.44 Om; xΣ = 2 · 3.51 + 12.61 = 19.63 Om;   Re ΔU 1 = 0.426 · 6.44 − (−0.835) · 19.63 + 0.125 · 1.09 − (−0.313) · 8.1 + 0.336 · 1.55 − (−0.579) · 3.51 = 24.36;   Im ΔU 1 = 0.426 · 19.63 + (−0, 835) · 6.44 + 0.125 · 8.1 + (−0.313) · 1.09 + 0.336 · 3.51 + (−0.579) · 1.55 = 3.948;   Re I x = 0.426 · 1.826 − (−0.835) · 5.589 + 0.125 · 0, 509 − (−0.313) · 2.286 + 0.336 · 0.4375 − (−0.579) = 6.95;   Im I x = 0.426 · 5.589 + (−0.835) · 1.826 + 0.125 · 2.286 + (−0.313) · 0.509 + 0.336 + (−0.579) · 0.437 = 1.107; ctg φk = (0.426 + 0.125 + 0)/(−0.835 − 0.313 − 0) = −0.48.

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Preliminary distance to the place of damage: L X = 9.784 +

1 30.49 − 24.36 + 3.948 · (−0.48) · = 11.19 km 6.95 − 1.07 · (−0.48)) · 0.403 0.3587

The value is 11.19 km < 12.563 km, therefore, the place of damage is within the considered section of the line. Specified calculation. The result can be clarified, taking into account the value of the zero sequence current in formula (17, Chap. 2), which feeds the SC on the opposite side of the system C2. In this case, the value of ctgϕk is determined from Eq. (15, Chap. 2), and the zero sequence current I C2 0 from C2 is determined by the formula in (73, Chap. 2). Since I C2 0 depends on the SC place, expression (17, Chap. 2) is a nonlinear equation. It can be represented in the form of canonical correlation, which is usually used to solve the equations of the steady-state mode of the electric network by mathematical approximation methods: L x = f (L x ),

(1)

where L x —line length before branch. Various methods for the approximate solution of nonlinear equations can be used, which include the methods of simple iteration, Seidel, Newton–Raphson, and others [32–35]. The choice of a method for solving nonlinear Eq. (1) is not a fundamental problem. To implement the specified calculation, the simplest method for solving nonlinear equations is used—the method of simple iterations. As an initial approximation, the preliminary distance to the place of damage L (0) X = 11.19 km is accepted. The convergence of the updated calculation was checked according to the following two conditions:    (i) (i−1)  (2)  L X − L X  ≤ ε; i ≤ 1000,

(3)

where i is the sequence number of the iteration; ε—accuracy of calculations. If condition (2) is not satisfied during 1000 iterations, then according to (3) it is considered that the process diverges. The calculation is performed with an accuracy of 0.0001 line length. For the studied line, the convergence accuracy of the iterative process is ε = 0.0012 km. The calculation gave the result at the 9-th iteration: L X = 11.30625 km with the accuracy of the solution εi = 0.00064 < ε.

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2.2 Damage of the “I-S” Overhead Line According to the record of the operational journal on 16/07/02 at 6:55 a single-phase SC took place on the “I-S” line. Phase B was damaged. The reason for disconnecting the overhead line is damage to the crosshead on the tower № 2. The distance to the place of damage was 0.15 km from the “I” substation. Description of the damaged line. The “I-S” overhead line is single-stranded, but has a magnetic connection on part of the route. The total length of the line from “I” substitution to “S” substitution is 5.3 km; the distance from the “I” substitution to the branches on the “P” substitution—2.33 km; branch length—1 km. Starting from the branch at the “P” substitution and to the “S” substitution the line is magnetically connected to the “P-S”. The neutrals of transformers at “P” substitution are closed. The line equivalent circuit is shown in Fig. 13. In the considered emergency shutdown record of the line, the full oscillogram of the emergency shutdown of which is shown in Fig. 14, two characteristic stages can be distinguished.

Fig. 13 Equivalent circuit of the “I-S” overhead line

Fig. 14 Complete oscillogram of emergency shutdown of “I-S” overhead line

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Fig. 15 Estimated EMP intervals of automatic operation mode in the analysis of emergency shutdown records of “I-S” overhead line

Stage 1: single-phase SC on the “I-S” overhead line. The first stage of the accident ends with the disconnection of the “IS” overhead line. SC points are fed from two sides of the line: from the “I” and “S” substations. Stage 2: single-phase SC on the “I-S” line due to unsuccessful ACR. The second stage ends with the repeated disconnection of the line “I-S” line. Automatic operation mode. Automatically it was detected: the damaged line— “I-S”, the type of SC—single-phase, the damaged phase—B, and calculated EMP interval shown in Fig. 15. As a result of this operating mode, one distance to the place of damage was obtained—the previous one. Refined determination of the distance was not carried out due to the lack of C2 system parameters in the database. The preliminary distance was 0.23 km (deviation +1.5% of the line length). Semi-automatic operation mode. In a semi-automatic mode of operation, the first and second stages of the accident were considered. 1st stage of the accident. The duration of the calculation interval of the EMP was 0.04 s, the delay from the initial moment of the SC was 0.07 s, which allowed to reliably abstract from the aperiodic component of the current (Fig. 16). The previous distance to the damage point was 0.225 km (deviation +1.4%). 2nd stage of the accident. As can be seen from the oscillogram, the current of the damaged phase at the beginning of the second stage of the accident has a clearly expressed non-sinusoidal character. Given this circumstance, the calculated EMP interval was chosen in the following way. The delay from the beginning of the second stage of the accident was 0.08 s, duration—0.04 s (Fig. 16). The distance to the damage site was 0.217 km (deviation +1.2%). Information operation mode. In this mode, the first and second stages of emergency shutdown of the line were considered. The calculated EMP intervals are selected similarly to the semi-automatic operation mode. The results of the information mode are shown in Table 3 for the first and second stages of the accident. The arguments of the vector values of currents and voltages are given in Table 3 relative to the voltage vector of the damaged phase “B”, directed along the real axis.

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Fig. 16 Estimated EMP interval of a semi-automatic operation mode at analyzing the emergency shutdown records of “IS” overhead line

Table 3 Result of emergency shutdown information mode of the “I-S” overhead line Mode parameters

Phase voltages

Phase currents

Ua

Ub

65.09

5.89

139.1

0

65

0.87

142.25

0

Uc

Ia

Ib

Ic

I stage emergency mode Effective value (kV, kA) Phase,

o

58.91

0.6

14.01

0.54

251.04

116.8

−59.44

124.69

II stage emergency mode Effective value (kV, kA) Phase,

o

68.16

0.63

13.93

0.53

253.96

119.29

−62.23

119.29

2.3 Damage of the “K-B” Overhead Line The considered damage was fixed by the Rekon-60BS hardware-software complex. The duration of the emergency shutdown recording is 0.75 s, of which the first 0.25 s falls on the pre-emergency mode. The registration frequency of instantaneous signal values is 900 Hz. According to the record of the operational journal on 26/06/02 at 5:17 a singlephase SC took place on the “K-B” overhead line. Phase C was damaged. The cause of the overhead line cut-off is damage to the crossarm at tower № 63. The distance to the damage zwxte was 11.6 km from the “K” substation. The “K-B” line is single-chain. The total length of the line from “K” substitution to “B” substitution is 24.5 km. The line equivalent circuit is shown in Fig. 17. A full oscillogram of the line emergency shutdown is shown in Fig. 18. In the considered line emergency shutdown record, only one characteristic stage can be

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Fig. 17 Equivalent circuit of the “K-B” line

Fig. 18 Complete oscillogram of emergency shutdown of “K-B” overhead line

distinguished. This is a single-phase SC on the “K-B” line, at the end of the stage it is eliminated by relay protection. Automatic operation mode. Automatically it was detected: the “K-B” line was damaged, the SC type—single-phase, the damaged phase—C and the calculated EMP interval shown in Fig. 19. As a result of this operating mode, only the preliminary distance was calculated, which amounted to 10.6 km (the deviation—4%). Semi-automatic operation mode. The duration of the calculation interval of the EMP is 0.04 s, the deviation from the initial moment of the SC is 0.09 s (Fig. 20). The preliminary distance to the place of damage was 11.26 km (error 1.5%). Information operation mode. The calculated EMP interval is similar to the interval selected in the semi-automatic operation mode. The result of the operation of the information mode is shown in Table 4. All the EMP are relative to the voltage vector of the damaged phase directed along the real axis.

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Fig. 19 Estimated EMP interval of automatic operation mode during analyzing the emergency shutdown record of “K-B” overhead line

Fig. 20 Estimated EMP interval for semi-automatic operation mode during analysis of emergency shutdown recording of “K-B” overhead line Table 4 Result of the information mode for the first stage of emergency shutdown of “K-B” overhead lines Mode parameters Effective value (kV, kA) Phase, o

Phase voltage of damaged line

Phase currents of damaged line

Ua

Ia

Ub 64.54

60.26

−104.92

139.24

Uc 17.22 0

Ib 0.068

−34.38

Ic 0.11

1.42

−50.03

−46.79

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References 1. Blinov, I., Zaitsev, I.O., Kuchanskyy, V.V.: Problems, methods and means of monitoring power losses in overhead transmission lines. In: Babak, V.P., Isaenko, V.M., Zaporozhets, A.O. (eds.) Systems, Decision and Control in Energy I. Studies in Systems, Decision and Control, pp. 123– 136. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-48583-2_8 2. Kuchanskyy, V.V., Malakhatka, D.O., Blinov, I.: Efficiency increase of open phase modes in bulk electrical networks. In: Zaporozhets, A., Artemchuk, V. (eds.) Systems, Decision and Control in Energy II. Studies in Systems, Decision and Control, pp. 31–48. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-69189-9_2 3. Ibrayev, N., Uyzbayeva, A.: Application of nonlinear surge arrestors for lightning protection of overhead power lines and modeling of overvoltage in dead-end substation. In: 4th International Conference on Power Engineering, Energy and Electrical Drives, 13–17 May 2013, Istanbul, Turkey, pp. 1156–1161. https://doi.org/10.1109/PowerEng.2013.6635774 4. Gebczyk, K., Chojnacki, A.L., Grakowski, L., Banasik, K.: Failures of insulators in low voltage overhead lines. In: 2019 Progress in Applied Electrical Engineering (PAEE), 17–21 June 2019, Koscielishko, Poland, pp. 1–5. https://doi.org/10.1109/PAEE.2019.8789000 5. Babak, S., Babak, V., Zaporozhets, A., Sverdlova, A.: Method of statistical spline functions for solving problems of data approximation and prediction of objects state. In: CEUR Workshop Proceedings, vol. 2353, pp. 810–821 (2019). https://ceur-ws.org/Vol-2353/paper64.pdf 6. Zaporozhets, A.: Overview of quadrocopters for energy and ecological monitoring. In: Babak, V., Isaienko, V., Zaporozhets, A. (eds.) Systems, Decision and Control in Energy I. Studies in Systems, Decision and Control, vol. 298, pp. 15–36 (2020). https://doi.org/10.1007/978-3030-48583-2_2 7. Zaporozhets, A., Kovtun, S., Dekusha, O.: System for monitoring the technical state of heating networks based on UAVs. In: Shakhovska, N., Medykovskyy, M. (eds.) Advances in Intelligent Systems and Computing IV. CCSIT 2019. Advances in Intelligent Systems and Computing, vol. 1080, pp. 935–950. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-33695-0_61 8. He, S., Yang, D., Li, W., Xia, Y., Tang, Y.: Detection and fault diagnosis of power transmission line in infrared image. In: 2015 IEEE International Conference on Cyber Technology in Automation, Control, and Intelligent Systems (CYBER), 8–12 June, Shenyang, China, pp. 431–435. https://doi.org/10.1109/CYBER.2015.7287976 9. Matikainen, L., Lehtomali, M., Ahokas, E., Hyyppa, J., Karjalainen, M., Jaakkola, A., Kukko, A., Heinonen, T.: Remote sensing methods for power line corridor surveys. ISPRS J. Photogramm. Remote. Sens. 119, 10–31 (2016). https://doi.org/10.1016/j.isprsjprs.2016. 04.011 10. Jalil, B., Pascali, M.A., Leone, G.R., Martinelli, M., Moroni, D., Salvetti, O., Berton, A.: Visible and infrared imaging based inspection of power installation. Pattern Recognit. Image Anal. 29, 35–41. https://doi.org/10.1134/S1054661819010140 11. Popovic, T., Kezunovic, M., Krstajic, B.: Smart grid data analytics for digital protective relay event recordings. Inf. Syst. Front. 17, 591–600 (2015). https://doi.org/10.1007/s10796-0139434-9 12. Mingottu, A., Peretto, L., Tinarelli, R., Zhang, J.: Use of COMTRADE fault current data to test inductive current transformers. In: 2019 II Workshop on Metrology for Industry 4.0 and IoT (MetroInd4.0&IoT), 4–6 June 2019, Naples, Italy, pp. 103–107. https://doi.org/10.1109/ METROI4.2019.8792871 13. Iegorov, O., Cheremisin, N., Savchenko, O., Diubko, S.: Improving the efficiency of managing the modes of distributive networks by automated monitoring the parameters of the mode and the environment in real time. Light. Eng. Power Eng. 1(54), 3–8 (2019). https://doi.org/10. 33042/2079-424X-2019-1-54-3-8 14. Wang, W., Huang, X., Tan, L., Guo, J., Liu, H.: Optimization design of an inductive energy harvesting device for wireless power supply system overhead high-voltage power lines. Energies 9(4), 242 (2016). https://doi.org/10.3390/en9040242

76

G. A. Senderovich et al.

15. Rezinkina, M., Rezinkin, O., Lytvynenko, S., Tomashevskyi, R.: Electromagnetic compatibility at UAVs usage for power transmission lines monitoring. In: 2019 IEEE 5th International Conference Actual Problems of Unmanned Aerial Vehicles Developments (APUAVD), 22– 24 Oct. 2019, Kyiv, Ukraine, pp. 157–160. https://doi.org/10.1109/APUAVD47061.2019.894 3932 16. Babak, V., Eremenko, V., Zaporozhets, A.: Research of diagnostics parameters of composite materials using Johnson distribution. Int. J. Comput. 18(4), 483–494 (2019) 17. Ukil, A., Yeap, Y.M., Satpathi, K.: Time-frequency domain analysis: wavelet-transform based fault detection. In: Fault Analysis and Protection System Design for DC Grids. Power Systems, pp. 223–242. Springer, Singapore (2020). https://doi.org/10.1007/978-981-15-2977-1_7 18. Ukil, A., Yeap, Y.M., Satpathi, K.: Design of experiment and fault studies. In: Fault Analysis and Protection System Design for DC Grids. Power Systems, pp. 325–354. Springer, Singapore (2020). https://doi.org/10.1007/978-981-15-2977-1_11 19. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Methods and models for information data analysis. In: Diagnostic Systems for Energy Equipments. Studies in Systems, Decision and Control, vol. 281, pp. 23–70 (2020). https://doi.org/10.1007/978-3030-44443-3_2 20. Shi, Y., Wang, L., Xie, R., Shi, Y., Li, H.: A 60-kW 3-kW/kg five-level T-type SiC PV inverter with 99.2% peak efficiency. IEEE Trans. Indus. Electron. 61(11), 9144–9154 (2017). https:// doi.org/10.1109/TIE.2017.2701762 21. Cao, L., Yang, J.: Linear circuit model of the three-phase insulated core transformer power supply. IEEE Trans. Nucl. Sci. 63(1), 288–296 (2016). https://doi.org/10.1109/TNS.2015.250 9026 22. Rezinkina, M., Rezinkin, O., Lytvynenko, S.: Simulation of electrical physical processes in electro-energetic systems at thunderstorm conditions. In: 2020 IEEE 15th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET), 25–29 Feb. 2020, Lviv-Slavske, Ukraine, pp. 322–326. https://doi.org/10.1109/TCS ET49122.2020.235448 23. Zaitsev, I.O., Kuchanskyy, V.V.: Corona discharge problem in extra high voltage transmission line. In: Zaporozhets, A., Artemchuk, V. (eds.) Systems, Decision and Control in Energy II. Studies in Systems, Decision and Control, pp. 3–30. Springer, Cham (2021). https://doi.org/ 10.1007/978-3-030-69189-9_1 24. Shevchenko, S., Danylchenko, D., Dryvetskyi, S., Minakova, K.: Influence of direct lightning strikes and lightning strikes near power lines with protected and non-insulated wires. In: 2018 IEEE 3rd International Conference on Intelligent Energy and Power Systems (IEPS), 10–14 Sept. 2018, Kharkiv, Ukraine, pp. 17–21. https://doi.org/10.1109/IEPS.2018.8559565 25. Plieva, M., Gavrina, O., Kabisov, A.: Analysis of technological damage at 110 kV substations in JSC IDGC of the North Caucasus-“Sevkavkazenergo”. In: 2019 International MultiConference on Industrial Engineering and Modern Technologies (FarEastCon), 1–4 Oct. 2019, Vladivostok, Russia, pp. 1–7. https://doi.org/10.1109/FarEastCon.2019.8934076 26. Ershov, A.M., Khlopova, A.V., Sidorov, A.I.: Technological violations on overhead lines with voltage of 10 Kv. In: 2020 International Conference on Industrial Engineering, Applications and Manufacturing (ICIEAM), 18–22 May 2020, Sochi, Russia, pp. 1–5. https://doi.org/10. 1109/ICIEAM48468.2020.9111899 27. Dovgalyuk, O., Omelianenko, H., Pirotti, A., Bondarenko, R., Syromyatnikova, T.: Reliability increase of the distribution electric networks operation in the implementation of the energy market in Ukraine. In: 2019 IEEE 6th International Conference on Energy Smart Systems (ESS), 17–19 April 2019, Kyiv, Ukraine, pp. 70–75. https://doi.org/10.1109/ESS.2019.876 4243 28. Maevsky, D., Besarab, O., Maevskaya, E., Berzan, V., Savieliev, A.: Ways and reserves of increasing the efficiency of electric power transmission lines. In: 2020 IEEE 15th International Conference on Advanced Trends in Radioelectronics, Telecommunications and Computer Engineering (TCSET), 25–29 Feb. 2020, Lviv-Slavske, Ukraine, pp. 618–622. https://doi.org/ 10.1109/TCSET49122.2020.235506

Experimental Studies of the Method for Determining Location …

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29. Breido, I., Kaverin, V., Abisheva, D., Kolychev, A.: Monitoring leakage currents of suspension insulators of high voltage overhead power lines. In: E3S Web of Conferences, vol. 140, p. 05010 (2019). https://doi.org/10.1051/e3sconf/201914005010 30. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Simulation and software for diagnostic systems. In: Diagnostic Systems for Energy Equipments. Studies in Systems, Decision and Control, vol. 281, pp. 71–90. Springer, Cham (2020). https://doi.org/ 10.1007/978-3-030-44443-3_3 31. Guillen, D., Paternina, M.R.A., Zamora, A., Ramirez, J.M., Idarraga, G.: Detection and classification of faults in transmission lines using the maximum wavelet singular value and Euclidean norm. IET Gener. Transm. Distrib. 9(15), 2294–2302 (2015). https://doi.org/10.1049/iet-gtd. 2014.1064 32. Mumtaz, F., Syed, M.H., Hosani, M.A., Zeineldin, H.H.: A simple and accurate approach to solve the power flow for balanced islanded microgrids. In: 2015 IEEE 15th International Conference on Environment and Electrical Engineering (EEEIC), 10–13 June 2015, Rome, Italy, pp. 1852–1856. https://doi.org/10.1109/EEEIC.2015.7165454 33. Kulworawanichpong, T.: Multi-train modeling and simulation integrated with traction power supply solver using simplified Newton-Raphson method. J. Modern Transport. 23, 241–251 (2015). https://doi.org/10.1007/s40534-015-0086-y 34. Montoya, O.D., Garrido, V.M., Gil-Gonzalez, W., Grisales-Norena, L.F.: Power flow analysis in DC grids: two alternative numerical methods. IEEE Trans. Circuits Syst. II Express Briefs 66(11), 1865–1869 (2019). https://doi.org/10.1109/TCSII.2019.2891640 35. Kecman, V.: Iterative k data algorithm for solving both the least squares SVM and the system of linear equations. In: SoutheastCon 2015, 9–12 April 2015, Fort Lauderdale, FL, USA, pp. 1–6. https://doi.org/10.1109/SECON.2015.7132930

Mathematical Models of Electric Fields of Electric Transmission Lines Marina M. Rezinkina , Yevgen I. Sokol , Artur O. Zaporozhets , Oleg G. Gryb , Ihor T. Karpaliuk , and Sergiy V. Shvets

Abstract The chapter contains analytical expressions for calculating strength of the electric and magnetic fields of the objects of the energy system. Such mathematical models are necessary to create a dependence of the behavior of these fields for various operating modes. It is separately noted that there cannot be ideal conditions for the propagation of electric and electromagnetic fields, therefore, the models provide for the absence and presence of objects in the zone of their operation are shown. Based on the complexity of mathematical dependencies in the description of the fields, assumptions are made that allow to simplify the development models. The chapter shows typical cases of the location of power lines. The complex values of the azimuthal and axial components of the electric field are recorded for them. Based on the calculation results, the electric field strength curves were constructed in the cross section perpendicular to the power transmission lines at different distances. Keywords Electric field strength · Magnetic field · Mathematical models · Power lines · Automated monitoring · UAVs

M. M. Rezinkina Department of Theoretical Electrical Engineering, National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine Y. I. Sokol National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine e-mail: [email protected] A. O. Zaporozhets (B) Department of Monitoring and Optimization of Thermophysical Processes, Institute of Engineering, Thermophysics of NAS of Ukraine, Kyiv, Ukraine e-mail: [email protected] O. G. Gryb · I. T. Karpaliuk · S. V. Shvets Department of Automation and Cybersecurity, National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 Y. I. Sokol and A. O. Zaporozhets (eds.), Control of Overhead Power Lines with Unmanned Aerial Vehicles (UAVs), Studies in Systems, Decision and Control 359, https://doi.org/10.1007/978-3-030-69752-5_5

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To develop a system for monitoring the safety of objects of the energy system, it is necessary to calculate the strengths of electric (EFs) and magnetic fields (MFs) both in the absence of objects in their area of operation and in their presence [1–5]. In the absence of objects in the sanitary zone of the OPLs, analytical methods for calculating electromagnetic fields can be used. Moreover, for UAV navigation, it is advisable to use information about the distribution of electric, not magnetic, field strengths, the levels of which depend on the OPLs current and are determined by their load. To assess the influence of the electromagnetic field on objects that are in their area of operation, numerical methods for calculating the electromagnetic field should be used.

1 Analytical Methods for EFs Modeling of OPLs At calculating the EF strength of OPLs using analytical methods, the following assumptions must be made [6]: • OPLs are considered parallel infinitely long cylinders, the charge of which is evenly distributed along their axes; • the voltage on the wires of OPLs varies according to a sinusoidal law with a frequency of 50 Hz; • the phase shift in time between the voltage of the wires of the OPLs is equal to 120°; • the surface of the earth is considered flat, and the earth itself with respect to air is absolutely electrically conductive and having zero potential; • the presence of supports, structures, technical and biological objects in the power transmission zone is not taken into account; • the presence of additional cables (lightning protection, compensation, etc.) is not taken into account; • it is allowed that the wires of the OPLs are in the air with a relative dielectric constant of 1; • the effective values of the EF are determined in a plane perpendicular to the direction of the wires of the OPLs, in the region of closest proximity of the wires to the ground. Under the assumptions made, the values of potentials, specific charges and EF strength can be written in symbolic form for complex values, and the EF is represented as the sum of the EFs of OPLs and their mirror reflections relative to the earth’s surface. Figure 1 shows typical cases of the location of power line wires [6–8]. If the wires of the OPLs are split, the equivalent radius of the wire is calculated by the formula [9]:  1 r = M · rϕ · a M−1 M ,

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81

Fig. 1 Typical locations of OPLs wires

where M—number of split wires of the phases of OPLs; r ϕ —radius of the cross section of the wires of the phases of OPLs, m; a—radius of the vicinity along which the wires of the split phases of the power transmission line are located, m. In the general case of an arbitrary arrangement of OPLs wires, the value of capacitance per unit length is calculated by the formula [10]: CS =

2π εe ε0 ,  √3 √ 3 r ·r ·r h 1 ·h 2 ·h 3 ln 2 · √3 r 12 ·r 23 ·r31  r 12

23

31

where h 1 · h 2 · h 3 – distance from the earth surface to each of the wires of OPLs; r12 · r23 · r31 —distance between the wires of OPLs; r12 · r23 · r31 —distance between the wires of OPLs and their mirror reflections. In the case of a vertical arrangement of OPL wires, as shown in Fig. 1a, the OPL capacity per unit length is calculated by the formula: CS =

 ln

√ 2D 3 2 r

·

 3

2π εr ε0 h min •(h min +D)•(h min +2D) (2h min +D)•(2h min +2D)•(2h min +3D)

,

where D—distance between the wires of the OPLs, m; h min —minimum distance of OPL wires to the ground (h min =min{h1, h2, h3}), m; εr —relative dielectric constant of the medium (air) in which the wires of OPLs; ε0 = 0.885 · 10−9 F/m – electric constant. In the case of a horizontal arrangement of OPL wires, as shown in Fig. 1 b, the OPL capacity per unit length is calculated by the formula: 2π εe ε0



CS = ln

r·

 3

2h min ·D

√ (4·h 2min +D2 )· h 2min +D2

.

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If the wires of the OPLs are located at the vertices of an equilateral triangle, as shown in Fig. 1c, the capacity of the OPLs per unit length is calculated by the formula: CS =

⎡ ln⎣ 2D r

 3 · √

2π εe ε0

⎤.

 √  h 2min · h min +D 23 ⎦  √ 2 2 (4·h 2min +D2 )· 2·h min +D 23 + D4

The complex values of the azimuthal and axial components of the EF of OPL strength at the point P(x P , y P ) (Fig. 1a) are calculated using the formulas [11]: Uφ · S . E˙ x (x P , y P ) = 2π εe ε0 ⎡   ⎤ x1 −x P P − (x −x x)12−x 2 − 2 2 + (x1 −x P ) +(h 1 −y P ) ⎢  1 P √+(h1 +yP )  ⎥ ⎢ ⎥ x2 −x P P ·⎢ + 21 + j 23 · (x −x x)22−x 2 − 2 2 + ⎥, +(h 2 +y P ) (x2 −x P ) +(h 2 −y P )  ⎦ ⎣  √   2 P P P + 21 − j 23 · (x −x x)32−x − (x −x x)32−x +(h +y )2 +(h −y )2 3

P

3

P

3

P

3

P

Uφ · S . E˙ y (x P , y P ) = 2π εe ε0 ⎡   ⎤ h 1 −y P P − (x −x h)12+y + + 2 (x1 −x P )2 +(h 1 −y P )2 ⎢  1 P √+(h1 +yP )  ⎥ ⎢ ⎥ h 2 +y P 1 3 P ·⎢ + 2 + j 2 · (x −x )2 +(h +y )2 + (x −x h)22−y 2 + ⎥. 2 P 2 P 2 P +(h 2 −y P )  ⎣    ⎦ √ h 3 −y P P + 21 − j 23 · (x −x h)32+y + +(h +y )2 (x −x )2 +(h −y )2 3

P

3

P

3

P

3

P

The EF effective value at point P(x P , y P ) is calculated by the formula: E(x P , y P ) =

     E˙ x (x P , y P )2 +  E˙ y (x P , y P )2 ,

    where  E˙ x (x P , y P ),  E˙ y (x P , y P )—moduli of complex values of the azimuthal and axial components of the EF of OPL voltage at a point. Comparison of the distributions of the actual values of the voltage of the EF of OPL calculated by the above formulas with the data [12], which are shown in Figs. 2a and 3a, as well as in Figs. 2b and Fig. 3b, respectively. Figures 4a and 4b show the same comparison with the [12] data.

Mathematical Models of Electric Fields …

83

Fig. 2 Calculated dependences of the EF strength in the cross section perpendicular to the OPL wires at a distance of 1 m from the earth’s surface

Fig. 3 Calculated dependences of the EF strength in the cross section perpendicular to the OPL wires at a distance of 3 m from the earth’s surface

Fig. 4 Calculated dependences of the EF strength in the cross section perpendicular to the OPL wires at a distance of 0.5, 1, 1.5 m from the earth’s surface

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References 1. Rezinkina, M.M.: Parameters of thin electromagnetic shields that provide a decrease in magnetic induction. Tech. Phys. 59, 155–161 (2014). https://doi.org/10.1134/S10637842140 20194 2. Rezynkina, M., Grinchenko, V.: Usage of electromagnetic shields for power frequency magnetic field mitigation in power industry. Tech. Electrodyn. 3, 15–16 (2012) 3. Zangi, H., Bretterklieber, T., Brasseur, G.: A feasibility study on autonomous online condition monitoring of high-voltage overhead power lines. IEEE Trans. Instrum. Meas. 58(5), 1789– 1796 (2009). https://doi.org/10.1109/TIM.2009.2012943 4. Wong, P.S., Janoska, M.A., Light, C., McCourt, R.W.: Long term magnetic field monitoring near power lines. IEEE Trans. Power Deliv. 12(2), 922–927 (1997) https://doi.org/10.1109/61. 584414 5. Zangi, H., Bretterklieber, T., Brasseur, G.: Energy harvesting for online condition monitoring of high voltage overhead power lines. In: 2008 IEEE Instrumentation and Measurement Technology Conference, 12–15 May, 2008, Victoria, BC, Canada, pp. 1364–1369. https://doi.org/ 10.1109/IMTC.2008.4547255 6. Sokol, Y.I., Rezinkina, M.M., Gryb, O.G., Vasilchenko, V.I., Zuev, A.A., Bortnikov, A.V., Sosina, E.V.: A method of complex automated monitoring of Ukrainian power energy system objects to increase its operation safety. Electr. Eng. Electromechanics 2, 65–70 (2016) https:// doi.org/10.20998/2074-272X.2016.2.12 7. Canete, F.J., Diez, L., Cortes, J.A., Entrambasaguas, J.T.: Broadband modelling of indoor power-line channels. IEEE Trans. Consum. Electron. 48(1), 175–183 (2002). https://doi.org/ 10.1109/TCE.2002.1010108 8. Yang, S., Franklin, G.A.: Modeling of overhead transmission lines with segmented shield wires for signal attenuation calculation in power line carrier systems. In: 2011 Proceedings of IEEE Southeastcon, 17–20 April 2011, Nashville, TN, USA, pp. 12–16. https://doi.org/10.1109/ SECON.2011.5752896 9. Taniguchi, Y., Baba, Y., Nagaoka, N., Ametani, A.: An improved thin wire representation for FDTD computations. IEEE Trans. Antennas Propag. 56(10), 3248–3252. https://doi.org/10. 1109/TAP.2008.929447 10. Vavilova, G.V., Gol’dshtein, A.E.: Instrument for in-process control of capacitance per unit length of an electrical wire. Meas. Tech. 61, 278–283 (2018) https://doi.org/10.1007/s11018018-1421-6 11. Uhriandt, D., Winkler, R.: DC column plasma kinetics in a longitudinal magnetic field. IEEE Trans. Plasma Sci. 29(3), 462–470 (2001). https://doi.org/10.1109/27.928944 12. Tzinevrakis, A.E., Tsanakas, D.K., Mimos, E.I.: Analytical calculation of the electric field produced by single-circuit power lines. IEEE Trans. Power Delivery 23(3), 1495–1505 (2008). https://doi.org/10.1109/TPWRD.2008.916748

Physical Modeling of Discharges in Long Air Gaps with the Presence of the Corona at the Tops of Grounded Objects Marina M. Rezinkina , Yevgen I. Sokol , Artur O. Zaporozhets , Oleg G. Gryb , Ihor T. Karpaliuk , and Sergiy V. Shvets

Abstract The chapter is devoted to the protection of equipment of electrical substations and power lines from the damaging effects of lightning. One of the negative phenomena on electrical equipment is the presence of a corona discharge. The chapter notes that corona discharge can be one of the stimulating conditions for lightning. A high-voltage experimental bench and a methodology for studying processes related to the advancement of the lightning leader channel to the ground and the “choice” of the place of impact are described. The results of experimental studies of breakdown processes in the presence of a corona discharge at the tops of grounded rods are presented. Lines of equal electric field strength are constructed for various cross sections with different rounding radii. Conclusions are drawn about the dependence of the influence on the probability of damage to the corresponding grounded electrode by a high-voltage discharge in the presence of a corona discharge on it and without corona discharge. Keywords Artificial lighting · Corona discharge · High-voltage discharge · Lightning leader channel · Simulation · Grounded rods · Electric field strength · Power lines

M. M. Rezinkina Department of Theoretical Electrical Engineering, National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine Y. I. Sokol National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine e-mail: [email protected] A. O. Zaporozhets (B) Department of Monitoring and Optimization of Thermophysical Processes, Institute of Engineering Thermophysics of NAS of Ukraine, Kyiv, Ukraine e-mail: [email protected] O. G. Gryb · I. T. Karpaliuk · S. V. Shvets Department of Automation and Cybersecurity, National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 Y. I. Sokol and A. O. Zaporozhets (eds.), Control of Overhead Power Lines with Unmanned Aerial Vehicles (UAVs), Studies in Systems, Decision and Control 359, https://doi.org/10.1007/978-3-030-69752-5_6

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1 Description of the High-Voltage Experimental Bench and Research Methodology Investigation of the processes associated with the advancement of the leader channel of lightning to the ground and the “choice” of the place of impact represent significant difficulties [1–5]. In this regard, in the high-voltage laboratories of the world, experimental studies of discharges in long air gaps are carried out. Although the ongoing electrophysical processes, in this case, as in experiments on the artificial initiation of lightning [6], differ from the processes occurring in natural lightning. Such studies [7–10], as a rule, form the basis of current standards for protection against lightning. In a number of publications (see, for example, [1, 2]), it is noted that the presence of a corona discharge on a grounded electrode if a leader channel of negative polarity approaches it affects the development conditions of an ascending counter leader. There are also publications describing studies of the dependence of the corona current on the configuration of a grounded electrode [11–13]. At choosing means of protection against lightning, it is important to assess the likelihood of lightning strikes on lightning rods and protected objects. Measuring the electrical parameters of the leader channel of lightning presents significant difficulties; therefore, physical and mathematical modeling of the electrophysical processes accompanying the lightning discharge has become widespread [14–17]. It is known that in some cases ascending leaders develop (see, for example, [14]) in thunderstorm conditions from grounded objects to the leader channel of lightning. This is due to the fact that most lightnings in the middle latitudes have a negative polarity [14]. The average EF required for the development of a leader channel of positive polarity (E + ) is E + ≈ 5 kV/cm, which is about 2 times less than the average EF required for the development of leader channels of negative polarity (E − ≈ 10 kV/cm) [18]. Therefore, the presence of electrons associated with the movement of leader channels of negative polarity can cause the development of a counter leader of positive polarity from grounded objects. It is known that the leader stage of the development of breakdown of long air gaps is preceded by the stage of development of the streamer corona [19]. A number of published sources describe experimental studies of the dependence of the corona current on the voltage of the EF, as well as the shape of the top of the grounded rod electrode which simulating a lightning rod, however, the influence of the corona intensity of the grounded electrode on the probability of a high-voltage spark entering it [15, 19] is not considered. According to some literature data [19], the presence of a corona on top of a grounded electrode reduces the probability of a counter leader developing from it. Based on this, a study was made of the effect of the magnitude of the corona current, which depends on the level of tension of the applied EF and the shape of the top of the grounded electrode, on the probability of its being hit by a high-voltage discharge. To study such processes, physical modeling was applied using a high-voltage stand, which provides influence to the gap “plane—rod on the plane” (the distance between the planes is 2.1 m) of a constant voltage of negative polarity |Ucon | ≤ 200 kV,

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Fig. 1 The scheme of the high-voltage stand and its photo: a electrical diagram of the stand; b diagram of a physical model for studying the effect of corona current on breakdown processes; c high-voltage hall. R1 = 490 k; RL = 22 k; R2 = 500 k; R3 = 1 G; R4 = 1 G; N—oscilloscope; F1—cut-off arrester; F2—protective arrester; pV—milliammeter; 1—pulse voltage generator; 2—voltage divider; 3—field forming system; 4—DC voltage generator; 5—ground plane; 6— potential plane; 7—grounding electrodes; 8—hole in the high voltage plane; 9—tops of grounded electrodes; 10—high voltage electrode; 11—ball measuring spark gap (layer diameter—1.5 m)

as well as the influence to the discharge gap “rod—rod” of the pulse voltage of negative polarity amplitude |Upul | ≤ 1 MV [20]. To explain the study processes, mathematical modeling of the distribution of the electron beam in the studied electrode systems was used. For simulation the processes of damage by a high-voltage discharge of grounded rod electrodes simulating lightning rods and protected objects, the high-voltage stand of the NTU “KhPI” was used. This stand contains a 10-meter-high Marx generator with 12 stages (Fig. 1) [21–23]. A physical model was created, including a DC voltage generator U con ≤ −200 kV. The voltage from the generator is supplied to the conducting plane, which has a size of 3 × 3 m. This plane is suspended at a height of D = 2.1 m above the grounded plane. The EF strength caused by the presence of a suspended plane imitates the conditions of a thunderstorm situation in which a corona discharge can occur on the tops of grounded electrodes. As a result of applying the voltage U con to the beginning of the high-voltage breakdown, in the space between 5 and 6 planes, the voltage of the EF appears, which is approximately equal to E 0 = U con /D. The presence of EF voltage can cause corona discharges at the vertices located on the ground plane of the electrodes of height h, the intensity of which is characterized by the current values of the measured corona current (I cor ).

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2 Results of Experimental Studies of Breakdown Processes in the Presence of Corona Discharge at the Tops of Grounded Rods In a number of technical programs, the rod electrodes are evaluated by replacing them with a thread with a charge uniformly distributed along its length or located in an external electron beam of an electrically conductive ellipsoid (for example, [14]). In this case, the tension of the electron beam can be represented in the form of analytical expressions. However, in order to take into account the mutual arrangement of the rods, the shape of their vertices, their potentials, the presence of an external EF, as well as ionized zones, it is necessary to use numerical calculation methods. For the considered systems, the application of the finite volume method [24, 25] is effective. With this approach, a grid is superimposed on the computational domain, in the nodes of which the values of the potentials and EF strengths are determined. The difference between this method and conventional finite-difference methods is that to obtain a solvable system, integrations of the initial equations are used over the volumes of unit cells into which the computational domain is divided. In this case, the conditions at the interfaces are automatically satisfied, which greatly simplifies the calculation of EFs in inhomogeneous media. For calculation the distribution of the EF around the high-voltage and grounded electrodes, simulating rod lightning rods in the conditions of a thunderstorm situation, the calculation system shown in Fig. 2 was used. To simulate the processes that occur when the leader channel of lightning approaches the ground, a pulse voltage generator was used (Fig. 1), which allows on to obtain voltage pulses of negative polarity with an amplitude of up to 1 MV, a front Fig. 2 Calculation system for modeling breakdown processes between high-voltage and grounded electrodes of the system: 1, 2—grounded and potential planes, respectively; 3—grounded electrode; 4—ML; 5—the top of the grounded electrode; 6—high voltage electrode, 7—hole in the potential electrode

Physical Modeling of Discharges in Long Air Gaps …

89

of 1.2 µs, and a pulse length of 0.5 · U pul at a pulse length of 50 µs. In the performed tests, the voltage U pul is applied to the high-voltage electrode, whose top is at a distance d from the vertices of the grounded electrodes. The high voltage electrode passes through a hole in a suspended plane that is energized by U con . The dimensions of this hole are 60 × 60 mm in order to avoid breakdown between the plane 6 and the electrode 10. As shown, measurements of the corona current carried out with the location of the grounded electrodes at a distance of d = 0.7 m and d = 0.14 m from the border of this hole in direction of the OX axis, its presence practically does not affect the magnitude of the corona current. In the course of the experiments, the influence degree of the intensity of the predown processes of the corona formation on the probability of damage to grounded electrodes by the leader of negative polarity was studied [15, 16, 18, 22, 23]. For this, it was used such a scheme of experiments. A suspended high voltage electrode (10, Fig. 1) was supplied from a pulsed voltage generator (PVG) (1, Fig. 1) by a pulsed voltage of negative polarity U pul . Two grounded electrodes were placed at an equal distance from the high-voltage electrode. Moreover, the top of one of them had the shape of a cone 0.14 m high, the base diameter 0.04 m, and the second—the shape of a sphere with a diameter of 0.045 m or 0.125 m. In the potential plane 6, suspended at a distance of 2.1 m above the grounded plane 5, a constant voltage U con was applied, the value which varied from 0 to −200 kV. During the experiments, the number of breakdowns on a grounded electrode with a peak in the form of a cone and with a rounded peak was recorded, and, in the case when a high voltage discharge hit both electrodes at the same time, the values of U con increased by one. Such experiments were carried out at various levels of constant voltage, which was applied to potential plane 6 (Fig. 1). Tables 1 and 2 show the results of experimental studies. The following notation is used in these tables: NP—number of breakdowns in a pointed electrode; NR— number of breakdowns in a rounded electrode; h—height of the grounded electrodes; P = NP/NR—relative frequency of a high-voltage discharge entering the electrode with a peak in the form of a cone; d—distance between the high voltage and grounded electrode; H—suspension height of the high voltage electrode; N—number of the series of experiments; N exp —the number of experiments in this series (NP + NR may not equal, since the discharge can fall into both grounded electrodes at the same time); [Pmin , Pmax ]—confidence interval for P for a given N exp for the binomial distribution [26, 27]; U pul —pulse voltage amplitude; U con —constant voltage supplied to the potential plane; 2R—rounding diameter of the grounded electrode; I c↑ —corona current of the grounded pointed electrode; I c~ —corona current of the grounded rounded electrode. The results shown in Table 1 correspond to the case when the distance between the high voltage and grounded electrodes (D = 0.44 m) is less than the maximum distance required for breakdown (d = 0.44 m, U pul = 0.84 MV). Under these conditions, breakdown occurs at the pulse front of the applied voltage: a typical waveform is shown in Fig. 3a. The measurements were carried out using a high-voltage divider (Fig. 1), the voltage from which was supplied to the oscilloscope through the optical

0.84

0.84

17/9 = 1.89

12

0.84

7/7 = 1

8

11/16 = 0.69

0.84

10/8 = 1.5

7

11

0.84

15/6 = 2.5

6

0.84

0.84

12/8 = 1.5

5

0.84

0.84

15/7 = 2.14

4

4/10 = 0.4

0.84

19/6 = 3.2

3

5/10 = 0.5

0.84

13/2 = 6.5

2

10

0.84

28/12 = 2.3

1

9

U pul , MV

P = NP/NR

N

200

160

110

0

110

0

120

0

200

160

110

0

U con , kV

18/0

0/0

10/1

0/0

6/0

0/0

18/0

0/0

mA/mA

I cor↑ / I cor~

0.045

0.045

0.045

0.045

0.045

0.045

0.045

0.045

0.125

0.125

0.125

0.125

2R, m

0.44

0.44

0.44

0.44

0.44

0.44

0.44

0.44

0.44

0.44

0.44

0.44

d, m

1.2

1.2

1.2

1.2

1

1

0.93

0.93

1.2

1.2

1.2

1.2

h, m

1.375

1.375

1.375

1.375

1.375

1.375

1.375

1.375

1.375

1.375

1.375

1.375

H, m

Table 1 Results for the distance between the high voltage and grounded electrodes (D = 0.44 m)

[1.4, 3.2]

[0.48, 0.84]

[0.3, 0.7]

[0.2, 0.65]

[0.9, 1.2]

[1.2, 2.5]

[1.6, 4.5]

[1.2, 2.4]

[1.5, 3]

[1.9, 7.1]

[2.6, 25]

[1.9, 9]

[Pmin , Pmax ]

22

22

19

17

19

17

19

17

22

22

14

14

N exp

Sample differences for experiment series 9–11 and 12 are not significant

Sample differences for experiment series 9–11 are not significant

Sample differences for experiment series 7–8 are not significant

Sample differences for experiment series 5–6 are not significant

Sample differences for experiment series 1–4 are not significant

Notes

90 M. M. Rezinkina et al.

U pul , MV

1.08

1.08

1.08

1.08

1.08

P = NP/NR

9/42 = 0.21

2/12 = 0.17

3/22 = 0.14

9/19 = 0.47

24/41 = 0.59

N

1

2

3

4

5

200

180

160

120

0

U con , kV

21/0

16/0

11.5/0

6/0

0/0

mA/mA

I cor↑ / I cor~

0.125

0.125

0.125

0.125

0.125

2R, m

1.01

1.01

1.01

1.01

1.01

d, m

0.93

0.93

0.93

0.93

0.93

h, m

1.085

1.085

1.085

1.085

1.085

H, m

Table 2 Results for the distance between the high voltage and grounded electrodes (D = 1.01 m)

[0.44, 0.70]

[0.33, 0.64]

[0.07, 0.33]

[0.04, 0.38]

[0.12, 0.36]

[Pmin , Pmax ]

48

26

24

13

45

N exp

Sample differences for experiment series 1–3 and 5 are not significant

Sample differences for experiment series 1–3 are not significant

Notes

Physical Modeling of Discharges in Long Air Gaps … 91

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Fig. 3 Typical waveforms during a discharge from a high voltage electrode to a ground rod electrode: a discharge at the front of the applied pulse (d = 0.44 m, U pul = 0.84 MV); b discharge at the decay of the applied pulse (d = 1.01 m, U pul = 1.08 MV)

path. To calibrate the measuring system, it was used a ball measuring spark gap with a layer diameter of 1.5 m (11, Fig. 1). At applying a high voltage with PVG between the layers of the spark gap, separated by a distance of 0.351 m, in 50% of cases a breakdown occurred. 50% of breakdown voltage was determined by standards, its value is 862.5 kV. The voltage scale for the waveform shown in Fig. 2a is 1.15 kV/mV, i.e., breakdown occurred at a voltage of about 750 kV. From the analysis of the experimental data, it follows that with a relatively large rounding radius of the grounded electrode (2R = 0.125 m) in the discharge mode, a pointed electrode is predominantly affected at the front of the pulse, regardless of the presence of the corona and the level of its prebreakdown current (Table 1, experiments No. 1–4). Figure 4 shows photographs of high voltage discharges during experiments. Figures 4a, b correspond to the ingress of a high-voltage discharge into an electrode with pointed and rounded vertices, respectively, and Fig. 4c into both electrodes. In order to explain the obtained experimental data on the breakdown of long air gaps, it was calculated the distributions of EFs in systems containing a grounded electrode with rounded or pointed tops, and also a high-voltage electrode of negative polarity located above it. Since the leader channel of lightning moves relatively slowly

Fig. 4 Photos of high voltage discharges

Physical Modeling of Discharges in Long Air Gaps …

93

to the ground (over a time of the order of 0.01 s [14]), and for the processes under consideration in physical modeling wave processes can be ignored, this calculation was performed in the quasi-stationary approximation. The intersection of the calculation system with the z = 0 plane is shown in Fig. 2. In the calculation, the presence of a conducting plane located at a distance of 2.1 m from the grounded plane, which is under the negative constant potential U con , as well as the hole through which the high-voltage rod electrode passes, was taken into account. As with the above mathematical modeling of prebreakdown electromagnetic processes of corona formation, these calculations were carried out using the finite volume method. Given that the system under study does not have axial symmetry, the calculation was carried out in a three-dimensional formulation [28, 29]. The boundary conditions in the calculations are also shown in Fig. 2. Taking into account the results of the EF simulation at applying a constant voltage U con to the potential plane (Fig. 2) in the case of high-voltage breakdown, it was assumed that, due to the presence of a corona, the rounding radius of the conical top of the grounded electrode is 1.5 mm. Figure 5 shows the results of calculating the zones in the cross section z = 0,

Fig. 5 Calculated lines of even voltage of the EF in the cross section z = 0 for the level E + = 5 kV/cm for the following system parameters: h = 1.2 m, d = 0.44 m, U pul = 750 kV (solid line—U con = −120 kV, dashed line—U con = −200 kV). 1—high voltage electrode, 2—grounded electrode; a top of the grounded electrode, which has the shape of a cone with a radius of rounding of 0.015 m due to the presence of the corona; b top of the grounded electrode, which has the shape of a sphere with a diameter of 0.045 m; c top of the grounded electrode, which has the shape of a sphere with a diameter of 0.125 m

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inside which the potential is greater than or equal to E + in the region between the vertices of the potential and grounded electrodes with vertices of different shapes. The solid lines correspond to the case of applying voltage U con = −120 kV to the potential plane 2 (Fig. 5), dashed lines—U con = −200 kV. An analysis of the calculated distributions of the EF voltage in the studied electrode systems showed that, at the discharge front of the pulse of the applied voltage, there is a continuous zone between the high-voltage and the grounded electrode, in which the EF module is at least E + = 5 kV/cm (Fig. 5). Thus, in this case, the conditions for the development of a spark from a grounded to a high-voltage electrode always exist if spark processes even arise, i.e., there is a region near the top of the grounded electrode where the condition for reaching E cr = 30 kV/cm is fulfilled. Using calculations and experiments, it was shown that at applying a constant voltage of up to 200 kV to the high-voltage electrode, the EF voltage at the top of the rounded electrode with 2R = 0.125 m is less than E cr , therefore, the prebreakdown corona does not appear on it. As shown by the calculations of the EF, when a pulse voltage U pul (t) is applied to the high-voltage negative electrode, whose maximum is max max 750 V (Fig. 3a), when U pul (t) increases to a level close to U pul at the about U pul top of a grounded electrode with a diameter of 0.125 m, E cr strength is reached, and development of a pulsed corona can begin around it. However, the time during which the EF voltage near the grounded electrode with such a peak becomes sufficient to start the breakdown processes is less than 250 ns, which is insufficient for the development of a pulsed corona. The EF voltage at the top of a grounded electrode with a conical apex substantially exceeds the EF voltage at the rounded apex of a grounded electrode [30]. Therefore, the processes of development of the pulsed corona begin near the conical vertex or even before the U pul application at U con > 0, or at significantly lower levels of U pul (t), and, therefore, much earlier than near the rounded top of the electrode. This can serve as an explanation for the fact that in most cases the discharge falls into a grounded electrode with a conical apex, and not into an electrode with a peak in the form of a hemisphere with a diameter of 0.125 cm (Table 1, experiments № 1–4), despite the fact that zone E ≥ E+ = 5 kV/cm for an electrode with a top in the form of a hemisphere with a diameter of 0.125 cm (Fig. 5c) is significantly wider than for an electrode with a pointed top (Fig. 5a). The breakdown processes were also studied using a grounded electrode with a smaller radius of rounding of the tops (2R = 0.045 m), at which the EF strength at its top is greater than E cr at |U con | > 120 kV and sufficient for the appearance of the corona. As the experiments show, the probability of a high-voltage discharge getting into a grounded electrode with such a top is greater than in a pointed electrode when a constant voltage is applied |U con | < 200 kV (Table 1, experiments No. 9–11). This effect can be explained by the fact that the continuous zone between the high-voltage and grounded electrodes, in which the EF modulus exceeds E + , at |U con | < 200 kV for a rounded electrode is much wider than for a sharpened one (Fig. 5a, b). As shown by the results of experiments, with the application |U con | = 200 kV, a significant increase in the number of defeats of the pointed electrode is observed

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(Table 1, experiment № 12), although the zones in which the propagation of the positive spark for the sharpened and rounded electrodes are almost identical (dashed curves in Fig. 5a, b). Thus, there is a certain threshold value of the voltage U con , at which qualitative changes occur in the processes associated with the formation and movement of the spark from the grounded to the high voltage electrode. Perhaps these processes become more intense due to a significant increase in the corona current, which means the number of avalanches that form a spark discharge towards the high-voltage electrode. The results shown in Table 2 correspond to discharges in the case where the distance between the high-voltage and grounded electrodes is close to the maximum value (d ~ 1.01 m), the minimum voltage required for breakdown U pul ~ 0.8–1 MV, and discharges as usually occur at the tail of the pulse. For the oscillogram shown in Fig. 3b, the scale is 1.4375 kV/mV, i.e., breakdown occurs at 862.5 kV. Calculations were made of the distribution of EF in the system with parameters: h = 0.93 m, H = 1.085 m, d = 1.01 m, U pul = −1.08 MV. As follows from these calculations, the voltage of the EF in the region connecting the vertices of the high-voltage and grounded electrodes is less than the critical level necessary for the development of a positive spark—E + . Therefore, in this case, the processes of breakdown significantly differ from the discharges considered above at the pulse front, when there is a continuous region with |E| ≥ E + . Thus, during discharges on the tail of pulses, a positive spark from a grounded electrode cannot develop, and a discharge develops from a high-voltage electrode. In this case, the EF at the top of the grounded electrode should have a significant effect on the movement of the spark from the high-voltage electrode, “attracting” it to itself. Thus, the probability of damage to the grounded electrode will be greater, the more it distorts the electron beam, increasing its tension near its peak. According to [31], in order to complete the process of approaching the spark from the negative high-voltage electrode and the ascending spark from the grounded electrode, it is necessary to have an EF with a strength (E bac ) of about E bac ≈ 2 kV/cm. Figure 6 in plane z = 0 shows the results of calculation of zones inside which the potential is greater than or equal to E bac = 2 kV/cm in the region between the vertices of the potential and grounded electrodes with vertices of different shapes. Moreover, Fig. 6a corresponds to a system containing a grounded electrode with a rounded top: 2R = 0.125 m, and Fig. 6b—to an electrode with a pointed top, rounded only because of the presence of a corona (2R = 0.015 m). In Fig. 6, the solid lines correspond to U con = −120 kV, the dashed lines correspond to U con = −200 kV. As can be seen from a comparison of Fig. 6a, b, with applying a constant voltage U con = −200 kV (dashed lines, Fig. 6), the distribution of the lines of the voltage level of the EF for grounded electrodes with pointed (Fig. 6a) and rounded tops (Fig. 6b) with a diameter of 2R = 0.125 m slightly. With applying a constant voltage U con = −120 kV (solid lines, Fig. 6), the distribution for grounded electrodes with rounded (Fig. 6a) and pointed (Fig. 6b) tops differs quite significantly: as can be seen from Fig. 6b, the voltage a high voltage electrode and a grounded electrode with a pointed top is weakened. This can serve as an explanation of the predominant damage caused by a high-voltage discharge of a grounded electrode with a rounded

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Fig. 6 Lines of equal EF strength, which were calculated at the level E bac = 2 kV/cm with system parameters: h = 0.93 m, d = 1.01 m, U pul = 862.5 kV; solid line—U con = −120 kV, dashed line—U con = −200 kV; 1—high-voltage electrode; 2—grounded electrode; a top of the grounded electrode has a conical shape with a rounding diameter of 0.015 m; b top of the grounded electrode has the shape of a sphere with a diameter of 0.125 m

top (Table 2, experiments number 1–3), since when |U con | < 180 kV, the voltage of the EF in the region between the vertices of the high-voltage and grounded electrodes with a rounded top significantly exceeds the EF voltage in the same zone for the case of a pointed electrode. Attached |U con | ~ 180–200 kV, the distribution of the EF voltage around the vertices of the rounded and sharpened electrodes differs less, therefore, the number of defeats of the pointed electrode increases significantly (Table 2, experiments number 4, 5). Thus, the physical and mathematical modeling of corona formation processes at the tops of grounded electrodes showed that the presence or absence of a corona, as well as its intensity, is proportional to the measured corona current, does not significantly affect the probability of a corresponding grounded electrode being struck by a high-voltage discharge, if the applied voltage level below critical level U con . This level corresponds to the level of the maximum module of the applied voltage, at which there is no correlation between the frequency of damage to the lightning rods and the value of the applied constant voltage. From experiments, the level of such a critical voltage is 160 kV. Then the level of critical strength for the systems under consideration is E cr ≤ U con /D = 160 × 103 /2.1 ≈ 0.8 × 105 V/m. This level of EF voltage is greater than typical EF strengths in lightning conditions: E light ≈

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103 –0.2 × 105 V/m [32]. It follows that in a thunderstorm setting, the intensity of pre-breakdown coronation will also not affect the likelihood of a lightning strike.

References 1. Rezinkina, M.M., Knyzyev, V.V., Kravchenko, V.I.: Statistical model of the lightning leader attraction to ground objects. Tech. Phys. 50, 1150–1157 (2005). https://doi.org/10.1134/1.205 1453 2. Rezinkina, M., Lytvynenko, S., Svetlichnaya, E., Rezinkin, O., Kubrik, B., Sosina, E.: Determination of the conditions of inception of an upward leader from grounded objects in thunderstorm conditions. In: 2018 IEEE 3rd International Conference on Intelligent Energy and Power Systems (IEPS), 10–14 Sept. 2018, Kharkiv, Ukraine, pp. 93–96. https://doi.org/10.1109/IEPS. 2018.8559533 3. Zaitsev, I.O., Kuchanskyy, V.V.: Corona discharge problem in extra high voltage transmission line. In: Zaporozhets, A., Artemchuk, V. (eds.) Systems, Decision and Control in Energy II. Studies in Systems, Decision and Control, pp. 3–30. Springer, Cham (2021). https://doi. org/10.1007/978-3-030-69189-9_1 4. Cooray, V., Rakov, V., Theethayi, N.: The lightning striking distance—revisited. J. Electrostat. 65(5–6), 296–306. https://doi.org/10.1016/j.elstat.2006.09.008 5. Jiang, R., Qie, X., Wang, C., Yang, J.: Propagating features of upward positive leaders in the initial stage of rocket-triggered lightning. Atmos. Res. 129–130, 90–96 (2013). https://doi.org/ 10.1016/j.atmosres.2012.09.005 6. Rakov, V.A.: Parameters of rocket-triggered lightning. Int. J. Plasma Environ. Sci. Technol. 4(1), 80–85 (2010). https://doi.org/10.34343/ijpest.2010.04.01.080 7. Baranov, M.I., Koliushko, G.M., Kravchenko, V.I., Rudakov, S.V.: A generator of aperiodic current pulses of artificial lightning with a rationed temporal form of 10 µs/350 µs with an amplitude of ±(100–200) kA. Instrum. Exp. Tech. 58, 745–750 (2015). https://doi.org/10.1134/ S0020441215060032 8. Wakasa, S.A., Nishimura, S., Shimizu, H., Matsukura, Y.: Does lightning destroy rocks?: results from a laboratory lightning experiment using an impulse high-current generator. Geomorphology 161–162, 110–114 (2012). https://doi.org/10.1016/j.geomorph.2012.04.005 9. Kuchanskyy, V., Zaitsev, I.O.: Corona discharge power losses measurement systems in extra high voltage transmissions lines. In: 2020 IEEE 7th International Conference on Energy Smart Systems (ESS), 2020, Kyiv, Ukraine, pp. 48–53. https://doi.org/10.1109/ess50319.2020.916 0088 10. Kalair, A., Abas, N., Khan, N.: Lightning interactions with humans and lifelines. J. Light. Res. 5, 11–28 (2013). https://doi.org/10.2174/1652803420131029001 11. Kiousis, K.N., Moronis, A.X., Fruh, W.G.: Positive corona discharge current in a thin wirecylinder electrode configuration. Int. J. Power Energy Syst 34(2), 43–50 (2014). https://doi. org/10.2316/Journal.203.2014.2.203-0032 12. Rezinkina, M., Rezinkin, O., D’Alessandro, F., Danyliuk, A., Guchenko, A., Lytvynenko, S.: Experimental and modelling study of the dependence of corona discharge on electrode geometry and ambient electric field. J. Electrostat. 87, 79–85. https://doi.org/10.1016/j.elstat. 2017.03.008 13. Kasdi, A.: Computation and measurement of corona current density and V-I characteristics in wires-to-plates electrostatic precipitator. J. Electrostat. 81, 1–8 (2016). https://doi.org/10.1016/ j.elstat.2016.02.005 14. Bazelyan, E.M., Raizer, Y.P. (2000). Lightning Physics and Lightning Protection. CRC Press 15. Rezinkina, M.M., Knyazyev, V.V., Kravchenko, V.I.: Mathematical description of leader channel propagation for selection of model experiment parameters and lightning guard systems. Tech. Phys. 52, 1006–1010 (2007). https://doi.org/10.1134/S1063784207080075

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16. Rezinkina, M., Rezinkin, O., Bean, C., Chalise, S., Glasty, J.: Statistical analysis for probable varying potential lightnings strokes to extended objects. Gaodianya Jishu/High Voltage Eng. 37(11), 2727–2732 (2011) 17. Rezinkina, M.M.: Technique for predicting the number of lightning strokes to extended objects. Tech. Phys. 53, 533–539 (2008). https://doi.org/10.1134/S1063784208050010 18. Rezinkina, M.M., Knyazev, V., Kravchenko, V.: Computation of the probability of lightning damage to ground objects. Tech. Phys. 52(1), 59–64. https://doi.org/10.1134/S10637842070 10100 19. D’Alessandro, F.: Experimental study of the effect of wind on positive and negative corona from a sharp point in a thunderstorm. J. Electrostat. 67(2–3), 482–487 (2009). https://doi.org/ 10.1016/j.elstat.2008.12.003 20. Niedbalski, J.: High-voltage multichannel rail gap switch triggered by corona discharges. Rev. Sci. Instrum. 74(7), 3520 (2003). https://doi.org/10.1063/1.1584081 21. Rezinkina, M.M., Rezinkin, O.L., Danyliuk, A.R., Revuckiy, V.I., Guchenko, A.N.: Physical modeling of electrical physical processesat long air gaps breakdown. Tech. Electrodyn. 1, 29–34 (2017) https://doi.org/10.15407/techned2017.01.029 22. Rezinkina, M., Rezinkin, O., D’Alessandro, F., Danyliuk, A., Lisachuk, G., Sosina, E., Svetlichnaya, E.: Influence of corona on strike probability of grounded electrodes by high voltage discharges. J. Electrostat. 83, 42–51 (2016). https://doi.org/10.1016/j.elstat.2016.07.005 23. Sokol, E.I., Rezinkina, M.M., Rezinkin, O.G., Gryb, O.G., Svetlichnaya, E.E.: Statistical model for determination of probability of lighting strokes to ground objects. Tech. Electrodyn. 2, 11–18 (2016) https://doi.org/10.15407/techned2016.02.011 24. Inan, U.S., Marshall, R.A.: Numerical Electromagnetics: The FDTD Method. Cambridge University Press (2011) 25. Wang, W.-G., Li, M., Hageman, S., Chien, C.L.: Electric-field-assisted switching in magnetic tunnel junctions. Nat. Mater. 11, 64–68 (2012). https://doi.org/10.1038/nmat3171 26. Babak, V., Eremenko, V., Zaporozhets, A.: Research of diagnostics parameters of composite materials using Johnson distribution. Int. J. Comput. 18(4), 483–494 (2019) 27. Hong, Y.: On computing the distribution function for the Poisson binomial distribution. Comput. Stat. Data Anal. 59, 41–51 (2013). https://doi.org/10.1016/j.csda.2012.10.006 28. Rezinkina, M.M.: The calculation of the penetration of a low-frequency three-dimensional electric field into heterogeneous weakly conducting objects. Elektrichestvo 8, 50–55 (2003) 29. Rezinkina, M.M.: Calculation of three-dimensional electric fields in systems with thin wires. Elektrichestvo 1, 44–49 (2005) 30. Rezinkina, M.M.: Simulation of electric fields in the presence of rods with rounded upper ends. Tech. Phys. 60, 337–343 (2015). https://doi.org/10.1134/S1063784215030238 31. Akyuz, M., Cooray, V.: The Franklin lightning conductor: conditions necessary for the initiation of a connecting leader. J. Electrostat. 51–52, 319–325 (2001). https://doi.org/10.1016/S03043886(01)00113-9 32. Marshall, T.C., McCarthy, M.P.: Electric field magnitudes and lightning initiation in thunderstorms. J. Geophys. Res. Atmos. 100(D4), 7097–7103 (1995). https://doi.org/10.1029/95J D00020

Mathematical Modeling of the Electromagnetic Processes of the Corona’s Formation During the Operation of Electric Power Facilities Marina M. Rezinkina , Yevgen I. Sokol , Artur O. Zaporozhets , Oleg G. Gryb , Ihor T. Karpaliuk , and Sergiy V. Shvets Abstract The chapter is devoted to the physical causes of the corona discharge in the elements of the electric power system. The most significant negative consequences of a corona discharge are the loss of electricity, as well as a decrease of its quality due to the appearance of higher harmonics. The cause of this discharge is the presence of an electric field with a significant gradient. Therefore, the section simulates electric fields in the presence of rods with rounded vertices and in the presence of curved interface surfaces. Conclusions are drawn on the influence of the geometry of rods with rounded vertices at the level of maximum electric field strength. A combined technique of mathematical modeling of electric field amplification on the rounded vertices of very long cylindrical rods is given. The calculation of the amplification of the electric field at the tops of long cylindrical rods is carried out. Keywords Power lines · Corona discharge · Simulation · Electricity · Higher harmonics · Electric field strength · Rods’ geometry · Long cylindrical rods The occurrence of corona discharges in the elements of the electric power system leads to a number of negative consequences [1–5]. The most significant of them are losses of electricity, as well as a decrease in its quality due to the appearance of M. M. Rezinkina Department of Theoretical Electrical Engineering, National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine Y. I. Sokol National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine A. O. Zaporozhets (B) Department of Monitoring and Optimization of Thermophysical Processes, Institute of Engineering Thermophysics of NAS of Ukraine, Kyiv, Ukraine e-mail: [email protected] O. G. Gryb · I. T. Karpaliuk · S. V. Shvets Department of Automation and Cybersecurity, National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 Y. I. Sokol and A. O. Zaporozhets (eds.), Control of Overhead Power Lines with Unmanned Aerial Vehicles (UAVs), Studies in Systems, Decision and Control 359, https://doi.org/10.1007/978-3-030-69752-5_7

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higher harmonics. The occurrence of high-frequency components of current, voltage, and EF strengths due to the presence of corona discharges also leads to the propagation of electromagnetic interference. These processes worsen the electromagnetic environment and the operating conditions of automation systems and control of electric power facilities, as well as digital and computer technology. The most common processes of this type include the formation of a corona discharge on the wires of OPL [6–8]. For eliminating this phenomenon, an increase in the cross section, as well as the splitting of the wires of the power lines, is used, as a result of which the EF strength levels decrease, and the coronation processes stop or their intensity decreases. A large number of studies over the past 100 years have been devoted to the issue of combating the processes of corona formation on wires of OPLs [9–13]. In developed countries (Western Europe, USA, Canada) this problem is practically solved: there a crown on the wires of OPLs arises only in adverse weather conditions. At the same time, the problem of improving the electromagnetic environment at such facilities of the united energy system of Ukraine as high-voltage substations is still relevant. The levels of EF in the territory of these objects are very significant and in some cases are sufficient for corona discharges to occur at the tops and sharp edges of the equipment elements of high-voltage substations. In order to investigate the dependence of the intensity of corona formation processes on the level of applied voltage and the shape of the vertices of grounded objects, mathematical and physical modeling can be used [14–19]. To unify the various shapes of the tops and edges of the elements of electric power equipment, such modeling can be carried out for the most unfavorable case: a ground electrode with vertices of different shapes—pointed in the form of a cone, and also rounded with radii of various sizes. Such a simulation makes it possible to determine the dependences of the corona current on the level of the applied EF voltage, as well as the geometry of the objects. The use of recommendations on the selection of parameters of the elements of the electric power system obtained as a result of such modeling will allow avoiding the occurrence of corona discharges at high-voltage substations or reducing the intensity of these processes.

1 Modeling of EFs in the Presence of Rods with Rounded Tops To reduce losses caused by the presence of corona discharges at electric power facilities, information is needed about the level of maximum tension at the tops of objects of various heights, that are in the zone of action of an external EF. In this case, it is necessary to determine how the ratio of the height and diameter of the rods simulating electric power objects affects the level of maximum strength. For this purpose, a mathematical model has been developed to describe EFs in the presence

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of curved media interfaces, the influence of the parameters of conductive rods with rounded vertices on the maximum levels of EF strength is analyzed [16]. As a rule, the vertices of the rods used in the technique are rounded. During calculating the EF in such systems, the problem of taking into account the curvature of the surface of the vertices of the rods arises. The application of the finite element method [20, 21] in this case does not fully solve the problem, since the using elements (for example, triangles) have angles, which means that in calculating the voltage of the EF on them it will be overestimated. Given that the main part of the rod is straightforward, it is advisable to use finite-difference methods [22, 23] for the calculation. However, it is known that the use of a rectangular computational grid to describe the EF in systems with curved surfaces leads to a significant increase in the calculated EF levels that actually occur. Moreover, the refinement of the computational grid only worsens the situation. In order to solve this problem, various techniques are used, for example, the representation of derivatives in the form of polynomials, rather than finite differences. However, this approach is associated with a complication of the problem, which means that the main advantages of finite-difference methods are lost: simplicity and the ability of taking into account nonlinear parameters of media. For calculating the EFs in such systems the so-called conformal schemes [24, 25] are also used, which provide for averaging the material parameters of the media over the volume of cells located at curved media interfaces. Using this approach for media, which material parameters differ by no more than 5–10 times, gives good results. If these values differ by several orders of magnitude (for example, when the leading rods are in the air), then using this method is ineffective.

1.1 Modeling of EFs in the Presence of Curved Interface Surfaces Let will consider cases where the distances, at which the electron beam changes, significantly exceed the characteristic sizes of the objects under consideration, therefore, the calculation can be performed in the quasi-stationary approximation. The equation describing the distribution of the electron beam will be obtained as follows. It will be write the Maxwell equations in the form: − → − → − → ∂D r ot H = γ E + ∂t

(1)

− → − → − → − → where H and E —strengths of MF and EF respectively; D = ε0 ε E ; ε0 = 0.885:10–11 F/m; ε—relative permittivity; γ—electrical conductivity. Further it will be used the finite volume method for numerical calculation. In this case, a rectangular computational grid is applied to the considered region, and the solved difference equation is obtained using conservation laws. It takes the divergence

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from both sides of Eq. (1), taking into account that in the steady state the second term on the right side is close to zero. Then it integrates the obtained equation over the volumes of unit cells into which the computational domain is divided and uses the substitution E = −grad(ϕ) (where ϕ—electric potential). Finally it can be written: 

∂ϕ · γ ds = 0 S ∂n

(2)

where S—sides of the parallelepiped that halve the distances between adjacent nodes; n—direction of the normal to the integration loop. Let dwell in more detail on the calculation of the EF in the vicinity of the rounded vertices of the rods. During using the finite volume method, the nodes in which the potentials are calculated, are usually located at the interfaces between the media. This allows to take into account the boundary conditions automatically, without writing them down with separate equations. It is proposed to use this approach for obtaining solutions for the EF strengths in the vicinity of curved media interfaces. It takes into account that the considered system contains conducting (rod) and non-conducting (air) medium [6, 11, 18]. Moreover, for the quasistationary case, the potential of the leading object can be considered constant. In order to simplify the calculations, it considers a two-dimensional axisymmetric system: for three-dimensional objects, the same approach is used. First, it was considered the case of a rectilinear media interface (Fig. 1a). The closed loop S, along which the integration of Eq. (2) is carried out, can be represented in the form of 4 segments perpendicular to the components of the EF strength: S 1r and S 2r (in the direction parallel to the Or axis), as well as S 1z and S 2z (in the direction parallel to axis Oz). This is possible because integrations (2) along the loops S1rB and S2rB adjacent to the interface between the media on both sides of it,

Fig. 1 Cell of the design scheme in the case of rectilinear (a) and curvilinear (b) media interfaces

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in this case, can be replaced by integration over S 1r and S 2r . Such a substitution is valid, since the regions over which the integration takes place are equal to each other: = S1rB S 1r = S2rB = S 2r = r (where r —spatial step in the direction of the Or axis), and the difference analogues of the z–x component of EF strengths on both sides − of the interface Ez+ i,j and Ezi,j are defined by the same expressions: as derivatives by a step back or a step forward during integrating over S1rB and S2rB or as central derivatives during integrating over S 1r and S 2r : E z i,+j = −

ϕi, j+1 − ϕi, j ϕi, j − ϕi, j−1 ∂ϕ ∂ϕ , E z i,−j = − , ≈− ≈− ∂z z ∂z z

(3)

where ϕ i,j —node potential (i, j); z —spatial step in the direction of the Oz axis. It can be written Eq. (2) in difference form as the sum of the integrals over the segments perpendicular to the Or and Oz axes, taking into account that γ e π/4 str str ; kZ = ; S , S —the lengths of 1, i f α > π/4 1, i f α < π/4 1z 1r cur the straight sections of the integration loop S; Si,j —length of the curved section S (Fig. 1b). In obtaining (11), (12), a simplified approach was used when it is considered cur along with the section S 1z or S 2z (in our case, this that for α > π/4 the curve Si,j str segment –S1z ) complements the part of the integration loop parallel to the Oz axis. cur is considered to be part of the integration loop, which If α < π/4, then the curve Si,j str ), complements together with the section S 1r or S 2r (in our case, this segment –S1r the part of the integration contour parallel to the Or axis. For this cell (Fig. 1b), the components I z2 and I r2 in (4) are determined by expressions (6) and (8), since when they are obtained, curved sections are not included in the integration loop. Finding cur str str , rectilinear segments S1z , S1r , and also di,j does the length of the curved sectionSi,j not present significant difficulties and can be performed, for example, numerically in the result of a smaller partition of cells lying at the interface. Moreover, thanks to the proposed approach, the border cells located inside the conductive medium can be virtually moved to its surface at determining of the EF in a non-conductive medium. Such a “movement” does not affect the determination of the EF inside the conducting medium, since the change in the distance between the internal nodes does not affect the calculated levels of electric potentials through significant differences between the conductivities of the conducting and non-conducting media. The same principles are used to obtain a numerical approximation (2) for a different arrangement of cells of the computational grid with respect to the interface [26, 27]. where kr =

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There are other calculation methods. So, with using finite-difference methods, the curvilinear interface of media can be replaced by a stepwise approximating surface. However, this leads to the appearance in the calculation of local zones with increased EF strength, which in reality do not exist. Moreover, the EF strength levels in the areas adjacent to the corners approximating the curved surface of the rectangular cells will be the greater, the smaller the step of the spatial computational grid. The distribution of the EF strength thus calculated on the surface of the conducting sphere is found in the uniform EF shown in Fig. 2 (solid curve 1). The calculation was carried out at a grid step  = 0.02·R (where R—radius of the sphere). For comparison, the same figure (curve 2) shows an analytical solution for the conducting sphere in a homogeneous external electron beam. As can be seen from a comparison of curves 1 and 2, the differences in the levels of EF strength for these two cases are quite large (up to 30%). As noted above, a number of techniques are used to eliminate this problem. All of them are rather cumbersome, which substantially complicates their practical implementation, in contrast to the described approach associated with the introduction of a non-orthogonal mesh or an increase in the approximation order of derivatives. Thus, the application of the proposed approach allows to take into account both the length of the curved sections of the interface and the distance between the nodes of the computational grid located in a non-conductive medium of the boundary and

Fig. 2 Calculated distributions of the EF strength on the surface of the conducting sphere of radius R, which is in a homogeneous external EF with a voltage of E 0 : 1—solution using stepwise approximation (solid curve); 2—analytical solution for |E|; 3—solution using the proposed method; 4—value of the EF strength obtained as the difference of analytical solutions for potentials at the nodes of the computational grid

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the interface. This is possible as a result of using the conservation law (in this case, a charge) to obtain solvable equations by integrating the original differential equations over the cell contours of the computational scheme, taking into account changes in the interface within each cell. Figure 2 (curve 3) shows the results of calculating the modulus of EF strength on the surface of a conducting sphere located in a uniform EF, obtained using the − → described approach. Levels | E | in the (i, j)th node are determined by the calculated value of the potentials in the nodes of the computational grid as averaged values over the cell volume as follows:   − → (12)  E  = Ez2i,j + Er2i,j where 

ϕi, j+1 − ϕi, j ϕi, j + z  ϕi+1, j − ϕi, j ϕi, j ≈ −0.5 · + r

E z i, j ≈ −0.5 · Eri, j

− ϕi, j−1 z − ϕi−1, j r

For nodes located in a conducting medium at a distance from the interface, less than the step of the computational grid, the corresponding components in (13) are calculated using (9), (10) taking into account the real distance to the interface. Figure 2 also shows the analytical solutions. Moreover, curve 2 is calculated − → using an analytical solution for | E |, and curve 4 represents the calculation results − → | E | in the form of the difference of analytical solutions for potentials in the nodes of the computational grid according to formulas of the form (9), (10), (13). As the calculations showed, the values of the EF strength obtained using the conformal scheme completely coincide with the data obtained using the stepwise approximation. This is explained by the fact that with such a large difference in the values of the electrical conductivity of a conducting and non-conducting medium (by 5–6 orders of magnitude or more), a several-fold decrease in the equivalent electrical conductivity of cells containing a conducting and non-conducting medium, which located at the interface, practically does not affect the calculated level potentials.

1.2 Calculation of EF in the Vicinity of an Electrically Conductive Cylindrical Rod The approaches described above allow to solve the following problem: calculation of the EF in the vicinity of the leading cylindrical rod of a lightning rod, which is located in an external vertically directed EF of intensity E 0 in a thunderstorm environment. Since the leader channels of lightning to the ground move relatively slowly (speed is about 104 –105 m/s [28]), the distances at which the EF changes

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Fig. 3 Investigated calculation system containing rods: 1—rod; 2—ground; 3, 4—PML layers

significantly exceed the characteristic sizes of lightning rods, so the calculation can be performed in a quasi-stationary approximation. A calculation system containing a conductive cylindrical rod with a rounded top (1), shown in Fig. 3. There are no analytical solutions for the EF strength in systems containing cylindrical rods with rounded tops. For finding the distribution of EF, the numerical finite volume method [29] was used. Given the axial symmetry of this system, a cylindrical coordinate system was used. It was assumed that the Oz axis coincides with the axis of the rod 1 and is perpendicular to the surface of the ground 2 (Fig. 3). The computational domain is bounded by a rectangle with sides z = 0, z = Z max , r = Rmax , r = 0 (Fig. 3) and is divided into elementary rectangular cells. In order to reduce the computational domain, the so-called perfectly-matched layers (PML) [30, 31] were introduced at its boundaries (see 3, 4 in Fig. 3). These layers perform an auxiliary function and are necessary to ensure a quick and nonreflective decay of a curved homogeneous electron beam caused by the presence of the studied objects during approaching the boundaries of the computational domain. Distribution of electronic components in PML layers is not taken into account in the calculation results. It is assumed that the electrical conductivity in such a layer of thickness d is a tensor, has different values in the directions of the coordinate axes Or, Oz and varies along the depth of the layer in accordance with the polynomial law. So, for the PML layer perpendicular to the Or axis (Fig. 3), the variation of the r-th—γrPML (r ) i z-o—γzPML (r ) components of the conductivity tensor in the direction of the Or axis is written as [31]:

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γrPML (r ) = γ0 · kr (r ); γzPML (r ) =

γ0 , kr (r )

where k r (r) = 1 + (k max −1)·(r/d)m ; k max —maximum value of k r at the outer boundary of the PML layer; m—exponent; γ 0 —value of the electrical conductivity of the medium adjacent to the inner boundary of the PML layer. The values of the components of the conductivity tensor in the PML layer, perpendicular to the Oz axis (Fig. 3), have the form: γzPML (z) = γ0 · k z (z); γrPML (z) =

γ0 , k z (z)

where k z (z) = 1 + (k max −1)·(z/d)m . The conductivity values in the zones of intersection of the PML layers are found by multiplying the corresponding components in each of the layers. The conditions at the boundaries of the computational domain (Fig. 3) were set as follows: ϕ = 0 on the surface of the ground (z = 0); ∂/∂n = 0 for R = 0 and for r = Rmax (at the outer boundary of the computational domain). In order to take into account that, at the upper boundary of the computational domain beyond the PML layer, the strength of the applied electron beam is E 0 , for z = zmax the condition ∂/∂n = kmax · E 0 was used. Comparison with the analytical solution for a sphere in a uniform EF showed that at using 10 PML layers with parameters m = 5, k max = 500 γ0 , the relative error in calculating the stresses and potentials of the EF is no more than 3%.

1.3 Influence of the Geometry of Rods with Rounded Tops at the Level of Maximum EF Strength In order to assess how the height of the lightning rod affects the possibility of developing a counter leader from it, the distribution of the electromagnetism at different heights of the rods was calculated. Preliminary calculations were carried out with a successive increase in the dimensions of the computational domain bounded by PML layers, in the directions of the coordinate axes, as well as with a decrease in the step of the computational grid. It was assumed that the solution adequately describes the distribution of the EF when the values of the stresses and potentials ceased to change with a sequential twofold decrease in the computational domain and an increase in its dimensions. From the analysis of the obtained data, the following conclusions were drawn: in order for the relative error not to exceed 3%, the dimensions of the computational domain in the radial direction should be no less than the height of the lightning rod (N), and in the vertical—1.2 times higher than H; the step of the computational grid should be no more than R/10 (where R—radius of the rounding of the rod).

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The case is considered when, in a thunderstorm situation, the leader channel of lightning is far from the lightning rod, and it can be assumed that a uniform EF is applied to it. According to literature data [32–34], for discharges in the systems “rod-plane”, there is the so-called critical radius of the rod. It is determined under the condition that the nature of the breakdown is identical in systems with rods whose radius is less than the critical radius: R ≤ Rcr , since under this condition the breakdown voltage does not depend on the value of R due to the presence of the corona. The case is considered when the radius of the lightning rod is equal to the critical radius—R = Rcr = 0.1 m [35], moreover, as a rule, the radius of lightning conductors does not exceed this value. Figure 4 shows the results of a numerical calculation of the dependence of E max — the maximum levels of EF strength in the vicinity of conductive rods that are in an external uniform field of strength E 0 (curve 1), from their height H. The calculation was carried out with the following parameters of the computational grid: r = z = R/10, zmax = 1.2·H, Rmax = H, H = var. Figure 4 also shows the results of engineering estimates for E max (curves 2–4). Curve 2 corresponds to the E max estimate in the form of the ratio of the potential of the EF at the height of the top of the grounded rod to its radius: E max = E 0 · H/R.

Fig. 4 Calculated dependences of the maximum EF strength in the vicinity of the conductive rod-lightning rod on its height H: 1—numerical calculation using the described approach; 2— dependence E max = E0 ·H/R; 3, 4—analytical solutions for an elongated leading ellipsoid at a distance from the vertex 1 = 0 (3) and 1 = R = 0.1 m (4), 5—polynomial approximation of curve 1

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Curves 3 and 4 are obtained as a result of using the analytical expression for the EF strength above an ellipsoid on its axis (for example, [36]) at a distance from its peak  = Rcr (curve 3) and  = 0 (curve 4). As can be seen from a comparison ∗ of curves 1–4, the value of E max = E max /E 0 , obtained on the basis of the actual shape of the lightning rod (cylinder whose top is rounded due to the presence of the crown), occupy an intermediate position with respect to the smallest (curve 3), which correspond to the distance from the rod equal to the critical radius Rcr , and the levels themselves, which correspond to a point on top of the equivalent ellipsoid (curve 4). In Fig. 4, curve 5 corresponds to the approximation of curve 1 by a polynomial: ∗ E max



H R

= 1.19 · 10

−5

 ·

H R

3

 − 0.0028 ×

H R

2

 H + 4.06 (13) + 0.85 R

Considering that the criterion for the start of a breakdown in air is the achievement of a breakdown strength E br = 30 kV/cm, from (14) it can be written the expression for the critical EF strength, upon application of which a corona discharge will develop at a tip of height H at its top:

E cr (H /R) ≈ 3 · 106 / 1.19 · 10(−5) · (H /R)3 − 0.0028 · (H /R)2 +0.85 · (H /R) + 4.06]W /m For the range of variation of the ratio of the height of the rod to the radius of curvature of its top within: 10 < H /R < 70, the calculated dependence curve E·(H/R) can be approximated by a simpler function: ∗ (H/R) ≈ 0.7 · (H/R) + 6.08 = 1.4/(D/H) + 6.08, E max

where D—diameter of the rounding of the rod’s top. From the last expression, it can be found the limiting value for the diameter of the rounding of the top of a rod object of height H, which is in the zone of action of an EF of intensity E 0 , a large increase of which will ensure that there is no corona on this object:  D > H · 1.4/ 3 · 106 /E 0 − 6.08 . The last expression can be used to estimate the diameter with which sharp edges should be rounded on rod objects of height H located in the area of the EF with a E 0 strength, so that corona discharges do not occur on them. The use of this formula for the previously described objects of physical and mathematical modeling showed the coincidence of forecasts on the appearance or not appearance of the corona.

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2 Combined Method of Mathematical Modeling of EF Amplification on Rounded Vertices of Very Long Cylindrical Rods At calculating the levels of increase in the EF strength at the vertices of very long rods (H/R ≥ 102 –103 , where H, R—the rod height and the radius of rounding of its top) using numerical methods, the problem arises of choosing the step of the computational grid that should be less than R. The imposition of grids with such a small step on the computational domain, whose dimensions must exceed H, leads to the fact that the number of unknowns of the system of equations N, which have being solved, is very large (N > 106 –108 ), which complicates these calculations or makes them impossible on modern computing tools. If the order of the system, which have being solved, is less than 106 , its calculation, as a rule, is not difficult.

2.1 Calculation of EF of a Cylindrical Long Rod with a Step of the Computational Grid Proportional to Its Radius It was performed a numerical calculation of the EF in the vicinity of a long conductive cylindrical rod with a rounded top, to which an external vertically directed EF with strength E 0 is applied. For such a calculation, it is advisable to use finite difference methods: in particular, the finite integration method. A feature of this method is the integration of the Maxwell equations over the elementary volumes V into which the computational domain is divided. As a result of the introduction of additional equations representing the conditions at the interfaces, it is not necessary, because they are performed automatically. Figure 5 shows a cross section of the investigated area by a plane passing through the axis of the rod. Since the rod has axial symmetry, a cylindrical coordinate system can be used and half of the studied area can be considered (the calculated area is shown in dashed lines in Fig. 5). The boundary conditions are shown in Fig. 5. It was assumed that the potentials and EF strengths depend only on the radial (r) and azimuthal (z) coordinates. It was divided the computational domain by a uniform grid with a step r i in the radial direction and with a step zj in the azimuthal direction. For each grid node, it was written the Maxwell equations provided that there are no free charges in the system: − → div D = 0

(14)

− → where D —electric induction, which is expressed in terms of the EF strength and − → − → electric potential: D = ε0 ε E = −ε0 ε · gradϕ; ε—relative dielectric constant; ε0 = 0.88510–11 F/m.

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Fig. 5 Settlement system

As a result of applying the integration operation over V-volumes of unit cells into which the computational domain is divided, and the use of the Ostrogradsky-Gauss theorem, it can be obtained (15) in the form:

∂ ds = 0, (15) ε· ∂n S where S—surface that covers the volume V; n—normal to the surface S. It was written the solving equation as a result of replacing in (16) the operations of differentiation with respect to space by differences in the values of the potentials ϕ i,j at the nodes of the computational grid:

r ϕi, j + z i, j = 0

(16)

where 

r ϕi, j = ϕi−1, j Bi, j − ϕi, j Bi, j + Ci, j + ϕi+1, j Ci, j ; 

z ϕi, j = ϕi, j−1 Di, j − ϕi, j Di, j + Fi, j + ϕi, j+1 Fi, j ;  Bi, j = (1/ri−1 )(ri − ri−1 /2) z j−1 εi−1, j−1 + z j εi−1, j /2;  Ci, j = (1/ri )(ri + ri /2) z j−1 εi, j−1 + z j εi, j /2;

  Di, j = 1/z j−1 (ri + ri /4)εi, j−1 ri + (ri − ri−1 /4)εi−1, j−1 ri−1 /2;

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Fig. 6 Calculation of EF of a long cylindrical rod with a step of the computational grid proportional to its length



  Fi, j = 1/z j (ri + ri /4)εi, j ri + (ri − ri−1 /4)εi−1, j ri−1 /2; i, j—indices related to the coordinates r and z, respectively. It was obtained a system of equations, with written the equation for each node of the computational grid, which was solved by the iterative method of variable directions using sweep. In order to reduce the computational domain, the so-called uniaxial PML were introduced at its boundaries. The dielectric constant of such layers is a tensor (parameter k max = 300), as a result of which the attenuation of the applied electron beam is quickly attenuated due to the presence of leading rods. Solid curves on Fig. 6 show the lines of the equal potential of the EF (in volts) obtained for this system (Fig. 5) using the described approach. The calculation was carried out with the following parameters: H = 10 m, R = 0.1 m, grid spacing R = r = z = R/10 = 0.01 m, Rmax = H, Z max = 2H, E 0 = 1 V/m. The EF strength is maximum where equipotential lines thicken (in this case, at the top of the rod). The use of the finite integration method made it possible to take into account the presence of a curved interface between media due to rounding of the rod’s top, which has the shape of a hemisphere with radius R.

2.2 Calculation of EF of a Long Cylindrical Rod with a Step of the Computational Grid Proportional to Its Length As noted above, at calculating the EF in systems with H/R ≥ 102 –103 , a problem arises related to the need to choose a partition step proportional to R. To solve it, an approach similar to that described in [37] can be used when, for a thin infinitely

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long rod, a change in the EF strength between the node of the computational grid, located on its surface, and the node adjacent to it in the radial direction does not occur according to a linear law, but is inversely proportional to the distance of the nodes to the axis of the rod. This approach has been expanded with respect to rods of finite length. In this case, the calculation of the EF can be done with a significantly larger step of dividing the computational domain, which is determined not by the radius of the rod, but by its length. In Fig. 6 dashed lines show the distribution of equipotentials calculated using this approach for a system with the same parameters as in the previous case, but with applying a computational grid, it has a 100-fold step in space: H = r = z = H/10 = 1 m. As can be seen from a comparison of the solid and dashed curves of Fig. 6, the potential levels for these two cases have a fairly good agreement. However, at the same time, it is impossible to calculate the levels of maximum EF strength at the tops of the rods through a large step of the computational grid.

2.3 Calculation of EF Amplification at the Tops of Long Cylindrical Rods From a comparison of the results of the above-described calculations of EF with a large and a small step in space, it follows that the potential levels at the nodes distant from the top of the rod at a distance H down as well as to the right and up (nodes A1 , A2 , A3 are shown in Fig. 7) coincide within 1.4–5.4% (Table 1). The values given in the table correspond to potential levels at nodes A1 , A2 , A3 (in volts), calculated at the grid step H . The calculation was carried out with the following parameters: H = var, R = 0.1 m, Rmax = H, Z max = 2H, E 0 = 1 V/m, the boundary conditions are shown in Fig. 5. In Table 1, the relative error shows the maximum difference between the potential levels calculated for small (R ) and large (L ) steps of the computational grid at nodes A1 , A2 , A3 , and is determined as follows:

Fig. 7 Calculated distribution of EF potentials: a around the top of the rod; b at the top of the rod

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Table 1 Value of potential levels in nodes A1 , A2 , A3 H/R 50

70

100

200

300

400

500

700

900

1000

A1

2.63

3.73

5.42

10.28

15

19.55

24

32.84

41.3

45.5

A2

5.29

6.8

8.8

16

22.65

29.1

35.4

48

60

66

A3

5.11

6.57

8.44

15.2

21.47

27.55

33.5

45.3

55.7

62.3

δ

0.054

0.05

0.014

   δ = max ϕ Ak ( H ) − ϕ Ak ( R )/ϕ Ak ( R ) , where ϕ Ak ( H ), ϕ Ak ( R )—values of potentials at nodes A1 , A2 , A3 , calculated at the grid spacing H and R , respectively; k = 1, 2, 3. Figure 7 shows the calculated potential distributions in the region around the top of the rod with H/R = 100 (other parameters are the same as for Fig. 6): thin lines correspond to the calculation results for a small grid step (R), and bold lines for large (H). Moreover, since δ at H/R = 100 is 0.014 (Table 1), the potential values at the nodes A1 , A2 , A3 in Fig. 7a practically coincide. Figure 7b shows the distribution of the EF strength modulus (E) with respect to the applied field strength (E 0 ), in the vicinity of the rod top, calculated for this case with a grid pitch R . Given that the potentials at distances H from the top of the rod in calculations with large and small steps differ by no more than 5.4%, it is possible to calculate the EF only in this area with a small spatial step (R ) and boundary conditions in the form of a linear approximation potential values at nodes A1 , A2 , A3 , calculated as described above using a large step of the computational grid (H ). The results of calculating the modulus of EF tension using this approach in the zone remote from the top of the rod at a distance of 0.1R in the azimuthal and 0.3R in the radial directions are in good agreement (the maximum differences do not exceed 3%) with the results of the calculation performed at superimposed on the entire computational domain (Fig. 5) a small step in space R . The calculations were carried out with the same system parameters as for Figs. 6 and 7. Using this approach, the maximum EF strengths at the vertices of rounded cylindrical rods (E max ) were calculated with respect to the applied field strength (E 0 ). The potentials at the nodes A1 , A2 , A3 used for these calculations are shown in Table 1. Moreover, it was considered rods with sufficiently large H/R ratios (Fig. 8). The curve shown in Fig. 8 is approximated using the polynomial: E max (L/R) ≈ 1.16 · 10−17 · (L/R)7 − 3.57 · 10−14 · (L/R)6 E0 + 4.17 · 10−11 · (L/R)5 − 2.35 · 10−8 · (L/R)4 + 6.97 · 10−6 · (L/R)3 − 1.29 · 10−3 · (L/R)2 + 0.78 · (L/R) + 4.665.

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Fig. 8 Calculated dependence of the ratio of the maximum EF strength at the tops of rounded cylindrical rods (E max ) to the applied field strength (E 0 ) on the value of H/R

References 1. Zhang, X., Heng, W., Yu, Z., Li, S., Zhu, J., Yan, K.: Comparison of styrene removal in air by positive and negative DC corona discharges. Int. J. Environ. Sci. Technol. 10, 1377–1382 (2013). https://doi.org/10.1007/s13762-012-0175-y 2. Zaitsev, I.O., Kuchanskyy, V.V.: Corona discharge problem in extra high voltage transmission line. In: Zaporozhets, A., Artemchuk, V. (eds.) Systems, Decision and Control in Energy II. Studies in Systems, Decision and Control, pp. 3–30. Springer, Cham (2021). https://doi. org/10.1007/978-3-030-69189-9_1 3. Yawootti, A., Intra, P., Tippayawong, N., Rattanadecho, P.: An experimental study of relative humidity and air flow effects on positive and negative corona discharges in a corona-needle charger. J. Electrostat. 77, 116–122 (2015). https://doi.org/10.1016/j.elstat.2015.07.011 4. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Principles of construction of systems for diagnosing the energy equipment. In: Diagnostic Systems For Energy Equipments. Studies in Systems, Decision and Control, pp. 1–22. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-44443-3_1 5. Wu, C., Xie, S., Qi, F., Li, B., Wan, J., He, J.: Effect of corona discharges on the inception of positive upward leader-streamer system. Int. J. Mod. Phys. B 27(28), 1350165 (2013). https:// doi.org/10.1142/S0217979213501658 6. Rezinkina, M.M., Rezinkin, O.L., Svetlichnaya, E.E.: Electric field in the vicinity of long thin conducting rods. Tech. Phys. 60, 1277–1283 (2015). https://doi.org/10.1134/S10637842150 90182 7. Thang, T.H., Baba, Y., Nagaoka, N., Ametani, A., Takami, J., Okabe, S., Rakov, V.A.: FDTD simulation of lightning surges on overhead wires in the presence of corona discharge. IEEE Trans. Electromag. Comp. 54(6), 1234–1243. https://doi.org/10.1109/TEMC.2012.2198919 8. He, W., Chen, X., Wan, B., Lan, L., Fu, W., Guo, H., Wen, X.: Characteristics of alternating current corona discharge pulses and its radio interference level in a coaxial wire-cylinder gap. IEEE Trans. Plasma Sci. 46(3), 598–605 (2018). https://doi.org/10.1109/TPS.2018.2801388 9. Kuchanskyy, V., Zaitsev, I.O.: Corona discharge power losses measurement systems in extra high voltage transmissions lines. In: 2020 IEEE 7th International Conference on Energy Smart Systems (ESS), 2020, Kyiv, Ukraine, pp. 48–53. https://doi.org/10.1109/ess50319.2020.916 0088 10. Rezinkina, M., Rezinkin, O., D’Alessandro, F., Danyliuk, A., Guchenko, A., Lytvynenko, S.: Experimental and modelling study of the dependence of corona discharge on electrode

Mathematical Modeling of the Electromagnetic Processes …

11.

12.

13. 14.

15.

16. 17.

18.

19.

20.

21.

22. 23. 24.

25.

26. 27.

28.

117

geometry and ambient electric field. J. Electrostat. 87, 79–85 (2017). https://doi.org/10.1016/ j.elstat.2017.03.008 Rezinkina, M.: Modelling of electric field strength amplification at the tips of thin conductive rods arrays. Progress Electromag. Res. M 88, 111–119 (2020). https://doi.org/10.2528/PIE RM19102702 Fews, A.P., Wilding, R.J., Keitch, P.A., Holden, N.K., Henshaw, D.L.: Modification of atmospheric DC fields by space charge from high-voltage power lines. Atmos. Res. 63(3–4), 271–279 (2002). https://doi.org/10.1016/S0169-8095(02)00041-8 Cooray, V.: Charge and voltage characteristics of corona discharges in a coaxial geometry. IEEE Trans. Dielectr. Electr. Insul. 7(6), 734–743 (2000). https://doi.org/10.1109/94.891983 Rezinkina, M., Rezinkin, O., Sokol, Y., Lytvynenko, S.: Mathematical modelling of the electric field in systems with conductive rods for lightning protection. In: 2018 IEEE 3rd International Conference on Intelligent Energy and Power Systems (IEPS), 10–14 Sept. 2018, Kharkiv, Ukraine, pp. 89–92. https://doi.org/10.1109/IEPS.2018.8559504 Sokol, Ye.I., Rezinkina, M.M., Sosina, E.V., Gryb, O.G.: Numerical computation of electric fields in presence of curvilinear interface between conductive and non-conductive media. Electr. Eng. Electromech. 1, 42–47 (2016). https://doi.org/10.20998/2074-272X.2016.1.08 Rezinkina, M.M.: Simulation of electric fields in the presence of rods with rounded upper ends. Tech. Phys. 60, 337–343 (2015). https://doi.org/10.1134/S1063784215030238 Rezinkina, M., Rezinkin, O., Chalise, S., Gupta, H., Bean, C.: Statistical modeling of the process of lightning attachment to extended objects. In: 2010 International Conference on High Voltage Engineering and Application, 11–14 Oct. 2010, New Orleans, LA, USA. https:// doi.org/10.1109/ICHVE.2010.5640852 Rezinkina, M.M., Rezinkin, O.L., Svetlichnaya, E.E., Sosina, E.V.: Combined calculation of electric field increase in the vicinity of tops of thin conducting rods. Tech. Electrodyn. 3, 10–16 (2015) Rezynkinam, M.M., Scherba, A.A., Grinchenko, V.S., Rezynkina, K.O.: Calculation choice of parameters of electromagnetic screens of complicated three-dimensional configuration. Tech. Electrodyn. 1, 10–16 (2012) Green, N.G., Ramos, A., Morgan, H.: Numerical solution of the dielectrophoretic and travelling wave forces for interdigitated electrode arrays using the finite element method. J. Electrostat. 56(2), 235–254 (2002). https://doi.org/10.1016/S0304-3886(02)00069-4 Oxborrow, M.: Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators. IEEE Trans. Microw. Theory Tech. 55(6), 1209– 1218 (2007). https://doi.org/10.1109/TMTT.2007.897850 Thomas, J.W.: Numerical Partial Differential Equations: Finite Difference Methods (vol. 22). Springer Science & Business Media (2013) Li, R., Chen, Z., Wu, W.: Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods. CRC Press (2000) Yu, W., Mittra, R.: A conformal FDTD algorithm for modeling perfectly conducting objects with curve-shaped surfaces and edges. Microwave Opt. Technol. Lett. 27(2), 136–138 (2000). https://doi.org/10.1002/1098-2760(20001020)27:2%3C136::AID-MOP16%3E3.0.CO;2-Q Yu, W., Mittra, R.: A conformal finite difference time domain technique for modeling curved dielectric surfaces. IEEE Microwave Wirel. Compon. Lett. 11(1), 25–27 (2001). https://doi. org/10.1109/7260.905957 Gaillard, N., Pinzelli, L., Gros-Jean, M.: In situ electric field simulation in metal/insulator/metal capacitors. Appl. Phys. Lett. 89, 133506 (2006). https://doi.org/10.1063/1.2357891 Kafafy, R., Lin, T., Wang, J.: Three-dimensional immersed finite element methods for electric field simulation in composite materials. Numer. Methods Eng. 64(7), 940–972 (2005). https:// doi.org/10.1002/nme.1401 Kong, X., Qie, X., Zhao, Y.: Characteristics of downward leader in a positive cloud-to-ground lightning flash observed by high-speed video camera and electric field changes. Geophys. Res. Lett. 35(5), L05816 (2008). https://doi.org/10.1029/2007GL032764

118

M. M. Rezinkina et al.

29. Eymard, R., Gallouet, T., Herbin, R.: Finite volume methods. Handbook Numer. Anal. 7, 713–1018 (2000). https://doi.org/10.1016/S1570-8659(00)07005-8 30. Hugonin, J.P., Lalanne, P.: Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization. J. Opt. Soc. Am. A 22(9), 1844–1849 (2005). https://doi.org/10. 1364/JOSAA.22.001844 31. Basu, U., Chopra, A.K.: Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation. Comput. Methods Appl. Mech. Eng. 192(11–12), 1337–1375 (2003). https://doi.org/10.1016/S0045-7825(02)00642-4 32. Rep’ev, A.G., Repin, P.B.: Dynamics of the optical emission from a high-voltage diffuse discharge in a rod-plane electrode system in atmospheric-pressure air. Plasma Phys. Rep. 32, 72–78 (2006). https://doi.org/10.1134/S1063780X06010077 33. Rep’ev, A.G., Repin, P.B., Pokrovskii, V.S.: Microstructure of the current channel of an atmospheric-pressure diffuse discharge in a rod-plane air gap. Tech. Phys. 52, 52–58 (2007). https://doi.org/10.1134/S1063784207010094 34. Mauseth, F., Nysveen, A., Isdstad, E.: Charging of dielectric barriers in rod-plane gaps. In: Proceedings of the 2004 IEEE International Conference on Solid Dielectrics, 2004. ICSD 2004, 5–9 July 2004, Toulouse, France, pp. 447–451. https://doi.org/10.1109/ICSD.2004.135 0387 35. Cooray, V.: Lightning Protection. The Institution of Engineering and Technology (2009) 36. Singh, J.P., Lele, P.P., Nettesheim, F., Wagner, N.J., Furst, E.M.: One- and two-dimensional assembly of colloidal ellipsoids in AC electric fields. Phys. Rev. E 79(5), 050401(R) (2009). https://doi.org/10.1103/PhysRevE.79.050401 37. Taflove, A., Hagness, S.C.: Computational electrodynamics: the finite-difference time-domain method. Artech house (2005)

Physical Modeling of the Electrophysical Processes of the Formation of the Corona During the Operation of Electric Power Facilities Marina M. Rezinkina , Yevgen I. Sokol , Artur O. Zaporozhets , Oleg G. Gryb , Ihor T. Karpaliuk , and Sergiy V. Shvets Abstract The chapter contains materials on elucidating the influence of the shape of the top of the rod electrode and the value of the applied electric field strength on the presence and value of the corona current on top of the grounded electrode. Physical modeling was used. The purpose of these experiments was to establish a correlation between the results of physical and mathematical modeling of electromagnetic processes during the development of corona discharges in order to verify the adequacy of the description by a mathematical model of the processes of the appearance of the corona, as well as their degree of intensity, due to the value of the corona current. The results of physical modeling of electromagnetic processes during the development of the corona on rod electrodes with tops of various shapes are presented, using which mathematical models of the processes of corona formation on the tops of rod electrodes are constructed. Keywords Corona discharge · Corona formation · Corona current · Electric field strength · Rod electrodes · Tops · Various shapes · Electric wires · Physical simulation · Mathematical modeling

M. M. Rezinkina Department of Theoretical Electrical Engineering, National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine e-mail: [email protected] Y. I. Sokol National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine e-mail: [email protected] A. O. Zaporozhets (B) Department of Monitoring and Optimization of Thermophysical Processes, Institute of Engineering Thermophysics of NAS of Ukraine, Kyiv, Ukraine e-mail: [email protected] O. G. Gryb · I. T. Karpaliuk · S. V. Shvets Department of Automation and Cybersecurity, National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 Y. I. Sokol and A. O. Zaporozhets (eds.), Control of Overhead Power Lines with Unmanned Aerial Vehicles (UAVs), Studies in Systems, Decision and Control 359, https://doi.org/10.1007/978-3-030-69752-5_8

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The literature describes experimental studies of the dependence of the corona current on the voltage of EF and the shape of the top of the grounded rod electrode, simulating a lightning rod, however, the task of mathematical modeling of these processes was not posed and the determination of the parameters of the electrodes ensuring the absence of a corona at certain EF strengths is applied [1–5]. In order to find out how the shape of the electrode rod’s top and the magnitude of the applied voltage of the electromagnet influence the presence and value of the corona current at the top of the grounded electrode, physical modeling can be used [6, 7]. It is advisable to interpret the obtained results using mathematical modeling of the distribution of the EF strengths in the considered systems. The purpose of these experiments was to establish a correlation between the results of physical and mathematical modeling of electromagnetic processes during the development of corona discharges in order to verify the adequacy of the description by a mathematical model of the processes of the appearance of the corona, as well as the degree of their intensity, due to the magnitude of the corona current [8].

1 Physical Modeling of Electromagnetic Processes During the Development of the Corona on Rod Electrodes with Vertices of Various Shapes For physical modeling of the processes of corona formation on the tops of grounded rod electrodes, the equipment of the NTU “KhPI” high-voltage bench was used [9, 10]. The layout of the high-voltage bench and its photo are shown in Fig. 1, Chap. 6 of this book. A constant EF that occurs when U con is applied between planes 5 and 6 (Fig. 1) simulates the conditions under which a corona discharge can occur through the EF on the tops of grounded electrodes. The value of the intensity of this EF is E 0 ≈ U con /D. The intensity of corona discharges can be estimated from the measured current value of the corona current I cor . Table 1 shows the measured levels of I cor at various values of U con and the heights of the grounded electrodes h. The experiments were carried out with various shapes of the vertices of the grounded electrodes: a cone with a height of 0.14 m and a base diameter of 0.04 m, as well as hemispheres with a diameter of 0.045 and 0.125 m. From the measurements it follows that the corona current for the grounded electrode with a vertex in the form of a sphere with a diameter 0.045 m is zero at h < 1.2 m. The corona current for a grounded electrode with a vertex in the form of a sphere with a diameter of 0.125 m is close to zero at all values of h. Figure 1 shows the measured dependences of I cor on U con at different heights of the grounded electrodes h. Curves 1–4 correspond to the corona current during making the top of the grounded electrode in the form of a cone with a height of 0.14 m and a base diameter of 0.04 m. EF calculations can be used to estimate corona current. It is assumed that the I cor value is proportional to the volume of regions in which E ≥ E cr . The relative corona

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Fig. 1 Experimentally obtained dependences of the corona current (Icor ) on the level of the applied constant voltage (U con ): 1−h = 1.2 m; 2−h = 1.15 m; 3−h = 1.05 m; 4−h = 0.93 m; 5−h = 1.2 m, 2R = 0.045 m (curves 1–4 correspond to the case when the top of the grounded electrode has the shape of a cone, curve 5 corresponds to the case when the top of the grounded electrode has the shape of a sphere with a diameter of 0.045 m)

Table 1 Measured corona current I cor [μA] as a function of U con and h |U con |, kV Cone h, m

120

160

180

200

0.93

10

21

30

38

1.2

21

39

55

70

h, m

120

160

180

200

1.2

0

5

7

12

Sphere (2R = 0.045 m)

200 200 current levels I ∗ = Icor (Ucon )/Icor (where I cor (U con ) is I cor at U con ; Icor is I cor at U con = 200 kV), as well as their comparison with experimental data were calculated in this way (Table 1) and presented in Fig. 2.

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Fig. 2 Comparison of the measured (solid curves) and calculated (dashed curves) values of the relative values of the corona current on the conical (1) and rounded tops (2) of grounded electrodes with a diameter of 0.045 m

2 Mathematical Modeling of Corona Formation Processes at the Tops of Rod Electrodes In order to explain the experimentally obtained dependences of the corona current on the shape of the vertices of the grounded electrodes and the values of the applied constant voltage, mathematical modeling of the distribution of the electromotive force in the considering systems was carried out. In engineering calculations, to evaluate the EF of a rod electrode, it is used a replacing it with a thread with a charge uniformly distributed along its length or representing it in the form of a leading ellipsoid located in the external EF [11]. In this case, the strength of the electron beam can be represented in the form of analytical expressions. However, in order to take into account the shape of the rods’ tops, numerical methods are needed. For the considered systems, the application of the finite volume method is effective. To calculate the distribution of the EF in the vicinity of lightning rods in the conditions of a pre-thunderstorm situation, the calculation system shown in Fig. 3 was used. Since the rod has axial symmetry, a cylindrical coordinate system was used for the numerical calculation. Half of the original system was considered, while its left boundary coincides with the axis of symmetry of rod 3 (Fig. 3). To reduce the dimensions of the computational domain during finding the distribution of EF, the so-called perfectly-matched layers (PML) [12, 13] are well aligned along its right boundary in the direction of the Or axis (Fig. 3). In such layers, the dielectric constant

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Fig. 3 Calculation system for modeling the processes of corona’s formation: 1, 2—grounded and potential planes, respectively; 3—ground electrode; 4—PML; 5—top of the grounded electrode; 6—high voltage electrode

is considered to be a tensor varying in depth of the layers in accordance with the polynomial law, and its components in the directions of the coordinate axes Oz and Or are different [14]. The boundary conditions in the calculations are shown in Fig. 3. To take into account the surface curvature of the rounded tops of the rods at applying a rectangular grid, the approach described in [14] was used. Figure 4 shows the calculated level of equal potential ϕ* = ϕ/U con in the vicinity of the grounded electrodes (3) with a height of h = 1.2 m, having different shapes of tops (Fig. 3). These electrodes are placed in the EF created by plane 2 under the constant potential U con and ground plane 1 (Fig. 3). As can be seen from Fig. 4, the greatest differences in the distributions of ϕ* occur in the vicinity of the tops of the grounded rods. For the considered cases of the tops of the grounded electrodes imitating lightning rods, zones were determined in which the EF voltage exceeded the critical level—E cr = 30 kV/cm, that is, the voltage at which the breakdown begins in air under normal conditions, and therefore, corona formation can occur. These zones for different U con correspond to different levels of excess of the voltage of the EF over the average strength in the gap between the grounded and potential planes E 0 (E 0 = Ucon /D): E∗ = |E|/E 0 . For each value of U con , the value of E* at which E cr is achieved can be determined from the relation: E∗ = E cr · D/Ucon .

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Fig. 4 Calculated distribution of lines of equal potential (ϕ* = ϕ/U con ) for the vertices of the grounded electrode in the form of a cone (a), a sphere with a diameter of 0.045 m (b) and a sphere with a diameter of 0.125 m (c): 1, 2—grounded and potential planes, respectively; 3—grounded rod electrode; 4—PML zone

So, at U con = 120 kV the level at which |E| ≥E cr , is equal to: E* = 52.5 at U con = 140 kV; E* = 45 at U con = 160 kV; E* = 40 at U con = 180 kV; E* = 35 at U con = 200 kV; E* = 31.5. Figure 5 shows the results of the calculation of zones inside which the potential is greater than or equal to E cr near the tops of the grounded electrodes with a top in the form of a cone (solid lines) and in the form of a sphere with a diameter of 0.045 m (dashed lines) at various levels of U con . These results correspond to the calculations ϕ* = const shown in Fig. 4a, b. The solid curves on Fig. 5 correspond to the case when the top of the grounded electrode has the shape of a sphere with a diameter of 0.045 m, the dashed curves corresponds to the case when the top of the grounded electrode has a cone shape. 1, 2 are zones in which the EF voltage exceeds critical at a certain level of voltage U con for the top of the grounded electrode in the form of a sphere and cone, respectively. Bold lines show the outer contours of the tops of the electrodes.

Fig. 5 Calculated distributions of lines of equal EF voltage at applied to the upper voltage plane: a U con = 120 kV; b U con = 60 kV; c U con = 180 kV; d U con = 200 kV

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As the calculations showed, at U con = 120 kV, the E cr level at the top of the grounded electrode with the top’s form of a sphere with a diameter of 0.045 m is not achieved, so the dashed curves in Fig. 5a are missing. As shown by the calculations of the EF for the case of a grounded electrode with a top in the sphere shape with a diameter of 0.125 m (Fig. 4c), the EF strength E cr = 30 kV/cm at its apex is not achieved: the maximum excess of the voltage at the top of the electrode with respect to the average voltage E 0 is E* = 21.6 (and the minimum level E* at which a corona can occur at U con = 200 kV is 32.5), so the corona should not ignite. This is also evidenced by the measurement results that did not record the presence of corona current for an electrode with such a top during applying to the potential plane |U con |=120–200 kV. The calculated results shown in Fig. 5 correlate well with the experimental data presented in Table 1: as the applied voltage U con grows, the area of the zone in which E ≥ E cr increases for both the electrode with a conical apex and the electrode with a spherical apex remains in the first case, much more than in the second.

References 1. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Simulation and software for diagnostic systems. In: Diagnostic Systems For Energy Equipments. Studies in Systems, Decision and Control, vol. 281, pp. 71–90. Springer, Cham (2020). https://doi.org/ 10.1007/978-3-030-44443-3_3 2. Babak, S., Babak, V., Zaporozhets, A., Sverdlova, A.: Method of statistical spline functions for solving problems of data approximation and prediction of objects state. In: CEUR Workshop Proceedings, vol. 2353, pp. 810–821 (2019). https://ceur-ws.org/Vol-2353/paper64.pdf 3. Filatov, I.E., Uvarin, V.V., Kuznetsov, D.L.: Estimation of qualitative and quantitative parameters of air cleaning by a pulsed corona discharge using multicomponent standard mixtures. Tech. Phys. 63, 680–688 (2018). https://doi.org/10.1134/S1063784218050079 4. Rezinkina, M., Rezinkin, O., D’Alessandro, F., Danyliuk, A., Lisachuk, G., Sosina, E., Svetlichnaya, E.: Influence of corona on strike probability of grounded electrodes by high voltage discharges. J. Electrostat. 83, 42–51 (2016). https://doi.org/10.1016/j.elstat.2016.07.005 5. Aleksandrov, N.L., Bazelyan, E.M., Carpenter, R.B., Drabkin, M.M., Raizer, Yu.P.: The effect of coronae on leader initiation and development under thunderstorm conditions and in long air gaps. J. Phys. D Appl. Phys. 34(22), 3256 (2001). https://doi.org/10.1088/0022-3727/34/ 22/309 6. Rezinkina, M.M., Sokol, E.I., Gryb, O.G., Bortnikov, A.V., Lytvynenko, S.A.: Calculation of electric field distribution in the vicinity of power transmission lines with towers and unmanned aerial vehicles presence. Tech. Electrodyn. 3, 3–9 (2018). https://doi.org/10.15407/techne d2018.03.003 7. Rezinkina, M.M.: Numerical calculation of the magnetic field and magnetic moment of ferromagnetic bodies with a complex spatial configuration. Tech. Phys. 54, 1092–1101 (2009). https://doi.org/10.1134/S1063784209080027 8. Salinas, E., Rezinkina, M.: Choice of parameters for passive shielding of power-frequency magnetic fields. Environmentalist 29(2), 135–140 (2009). https://doi.org/10.1007/s10669-0089208-y 9. Baranov, M.I., Koliushko, G.M., Kravchenko, V.I., Nedzel’skii, O.S., Dnyshchenko, V.N.: A current generator of the artificial lightning for full-scale tests of engineering objects. Instrum. Exp. Tech. 51, 401–405 (2008). https://doi.org/10.1134/S0020441208030123

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10. Karakurkchi, A., Sakhnenko, M., Ved, M., Yermolenko, I., Pavlenko, S., Yevsieiev, V., Pavlov, Ya., Yemanov, V.: Determining features of application of functional electrochemical coatings in technologies of surface treatment. East.-Eur. J. Enterp. Technol. 3, 12(99), 29–38 (2019). https://doi.org/10.15587/1729-4061.2019.171787 11. Bazelyan, E.M., Raizer, Y.P.: Lightning Physics and Lightning Protection. CRC Press (2000) 12. Hugonin, J.P., Lalanne, P.: Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization. J. Opt. Soc. Am. A 22(9), 1844–1849 (2005). https://doi.org/10. 1364/JOSAA.22.001844 13. Basu, U., Chopra, A.K.: Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation. Comput. Methods Appl. Mech. Eng. 192(11–12), 1337–1375 (2003). https://doi.org/10.1016/S0045-7825(02)00642-4 14. Rezinkina, M.M.: Simulation of electric fields in the presence of rods with rounded upper ends. Tech. Phys. 60, 337–343 (2015). https://doi.org/10.1134/S1063784215030238

Acoustic Diagnostics for Determining the Appearance of Corona Discharge Oleg G. Gryb , Ihor T. Karpaliuk , Artur O. Zaporozhets , Sergiy V. Shvets , and Natalia V. Rudevich

Abstract The chapter shows the dependence of quality indicators on the corona discharge on conductive elements. The dependence of the presence of higher harmonics in the electric network and corona discharge is noted. The direction of the search for the location in the space of the corona discharge was selected for the possibility of combating it. The direct determination of the presence of a corona discharge by electrical parameters (current shape and voltage shape) is problematic for making measurements directly on the lines. Therefore, other types of corona discharge factors were analyzed. Corona discharge consumes energy, which is converted into other forms of energy (or chemical compounds), and remains in the space surrounding them. The authors drew attention to acoustic noise in electrical installations and associated them with corona discharge. A laboratory bench was created and the connection of acoustic noise and the presence of corona discharge were confirmed. The chapter presents the results of the experiment. Acoustic noises were decomposed according to the Fourier transform and the spectral groups were determined that are characteristic of a corona discharge depending on the voltage at the rods. A three-dimensional acoustic model of the corona discharge was constructed, which made it possible to create an algorithm for searching for the source of the corona discharge. Keywords Corona discharge · Acoustic noises · Acoustic measurements · Fourier transform · Physical models · Simulation · Analysis · Power lines

O. G. Gryb · I. T. Karpaliuk · S. V. Shvets · N. V. Rudevich Department of Automation and Cybersecurity, National Technical University “Kharkiv Polytechnic Institute”, Kharkiv, Ukraine e-mail: [email protected] A. O. Zaporozhets (B) Department of Monitoring and Optimization of Thermophysical Processes, Institute of Engineering Thermophysics of NAS of Ukraine, Kyiv, Ukraine e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 Y. I. Sokol and A. O. Zaporozhets (eds.), Control of Overhead Power Lines with Unmanned Aerial Vehicles (UAVs), Studies in Systems, Decision and Control 359, https://doi.org/10.1007/978-3-030-69752-5_9

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1 Relationship of the Qualitative Parameters of Electrical Energy and Corona Discharge Qualitative parameters of electricity in the presence of disturbing factors in the network due to the occurrence of a corona discharge will change over time [1–3]. As shown in a number of works, a corona discharge consumes energy, that is, a corona discharge current arises, which also has a reactive component [4–11]. As a result, a reactive current arises in the network, which changes the qualitative parameters of electric current and voltage. In addition, the frequency component of the corona discharge includes not only the first harmonic. Many researchers have documented that in the corona discharge on high-voltage lines there are current components of the third and fifth harmonics [12–14]. The currents at these harmonics have a reactive component. Thus, the source of voltage and current distortion can also be a corona discharge, which introduces a reactive component, and leads to the presence of higher harmonics. One of the components of low quality electric energy is the presence of higher harmonics in the electric network [15, 16]. The sources of higher harmonics are traction substations of the main electric transport and other nonlinear and heterogeneous structural elements of electric network facilities. In addition, higher harmonics increase the likelihood of a corona discharge in characteristic places of electric networks [17, 18]. The corona discharge itself distorts the voltage and current curves in electrical networks. These two factors are connected by positive feedback: the appearance of higher harmonics leads to the appearance of a corona discharge, and the corona discharge, in turn, leads to the appearance of higher harmonics. Therefore, the issue of determining the location of a corona discharge on highvoltage lines is an actual direction in the development of diagnostic equipment in the world [19–23]. The direct determination of the presence of a corona discharge by electrical parameters (current shape and voltage shape) is problematic for making measurements directly on the lines. Therefore, manufacturing equipment for operation at significant voltages is not an easy task. In addition, the value of the currents from the corona discharge in a local place is insignificant and it is not always possible to carry out measurements with high accuracy to identify only corona discharge currents. Performing such measurements still requires the use of high precision instruments, and such measurements cannot be constant over time. There are always a lot of disturbances in the network, which leads to unpredictable results. It is also necessary to take into account the performance of such measurements and the manufacture of devices for them are quite expensive. Therefore, developers are moving towards determining the presence of a corona discharge by other (indirect) parameters. As shown in chapter “Physical Modeling of the Electrophysical Processes of the Formation of the Corona During the Operation of Electric Power Facilities”, a corona discharge consumes energy, which is converted into other forms of energy (or chemical compounds), and remains in the surrounding space. And such transformations include the energy of electromagnetic radiation, chemical transformation and mechanical work.

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Electromagnetic radiation occurs in the thermal range, visible and ultraviolet. Based on such radiation, many devices have been developed for detecting the monitoring of the presence of a corona discharge [24–26]. Such optical fixation methods have the advantage of being able to identify a specific location (localization) of the corona discharge. But such methods have a number of disadvantages, which include: • background flare (at which it is impossible to distinguish optical bursts from the corona, for example, solar radiation), • minor corona energy bursts (sensitive optical sensors), • presence of optical interference (most corona discharges occur during rain, fog, or snowfall), • determination only at short distances (up to 10 m). Chemical compounds, forming from a corona discharge, have certain characteristics. Therefore, corona discharge is used to obtain chemical compounds in the respective reactors. But diagnosing the presence of a corona discharge on the line by such a parameter does not make sense [27–30]. The next method is based on the fact that during the corona discharge there is also a local expansion of atmospheric air, which creates acoustic vibrations. The method was based on the registration and analysis of acoustic vibrations, according to which a series of experiments was performed to diagnose the presence of a corona discharge. The method itself is called spectral-acoustic. The measurements were performed on the acoustic vibrations of atmospheric air, followed by the decomposition of the acoustic signal into spectrum components.

2 Acoustic Noises in Electrical Installations and Corona Discharge An attempt to diagnose electrical equipment by its acoustic noise was carried out by many authors [31–37]. But basically such a diagnosis is associated with the identification of mechanical damage or is associated with the mechanical parameters of the electrical device itself—as example, acoustic diagnostics of an electric motor. Deterioration in the parameters of its bearings leads to additional noise. And for different types of defects and various types of engines, the nature of the noise will be different. But even on electric machines without mechanical mechanisms (for example, power transformers), additional noise is also possible, which may be a consequence of pre-emergency conditions [38–40]. But in this work, an analysis of acoustic vibrations is carried out, the source of which is not equipment, but an electrical phenomenon. There are many types of equipment and creating reference models of acoustic fields of the corresponding type of equipment is a very voluminous task. There are a lot of options for building libraries of standard noise of equipment, and practice has shown that the nature of noise can even depend on the production batch of equipment. Therefore, not a

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mechanical source, but a corona discharge [41, 42] was considered as a source of acoustic noise. Corona discharge has a fairly stable characteristic noise. Therefore, an assumption was made about the possibility of diagnosing the presence of a corona discharge by acoustic noise. According to the work of researchers about the phenomenon of corona discharge [4–6, 43, 44], it can be concluded that the source of acoustic vibrations in the corona discharge can be the front of movement of ions, expansion of air due to streamer breakdowns, expansion/contraction of air due to chemical reactions taking place in the corona discharge. The discharge can be in several spatial manifestations: in the form of a point discharge or in the form of a group discharge (cover). The registration of acoustic vibrations from the corona discharge is carried out at a distance that is several meters or tens of meters. The dimensions of the acoustic emission zone by the corona discharge are much smaller than the indicated distances; therefore, for distances greater than the main radius of the corona discharge losses (20 cm for the corona on the 500 kV line [45]), the discharge itself in space can be considered as a point source, and in cases of discharge in the form of the cover can be considered as linear. Due to such a simplification, two options will be considered below as an acoustic source of the corona discharge: a point source and a linear source. Accordingly, calculations of the propagation of acoustic waves for corona sources are calculated as for acoustic sources of the indicated forms.

3 Experimental Studies of the Acoustic Component of the Corona Discharge Measurements of the acoustic component from the corona discharge were carried out at the National Technical University “NTU “KhPI””. A corona discharge was obtained for a P-3 type insulator (with a breakdown limit of 35 kV). A high voltage was obtained from a step-up transformer (150,000/100 V), which was connected to a 220 V network through a laboratory autotransformer. The voltage at the high-voltage transformer was fixed with a voltmeter mounted on the low-voltage winding. The voltmeter data were converted to voltage on the high voltage side. The calibration of the ratio of the voltage on the high-voltage coil to low-voltage was carried out according to the discharge on the ball gap. Acoustic measurements were carried out by a group of instruments recording acoustic vibrations. The main device was a UMIK-1 microphone with a linear frequency response, auxiliary were small-sized voice recorders from SonyWalkmanNWZ-B173F and Transcend and others. The parameters of the UMIK-1 microphone are shown in Table 1. The parameters of the TranscendT Sonic 330 are shown in Table 2.

Acoustic Diagnostics for Determining the Appearance … Table 1 UMIK-1 microphone parameters

Table 2 TranscendT Sonic 330 microphone parameters

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№

Parameter

1

Capsule type

Electret

2

Frequency response

20 Hz–20 kHz

3

Frequency response deviation

±1 dB

4

Resolution and sample rate

24 bit, 44.1 or 48 kHz

5

Noise output

−74 dBFS

6

Amplification

133 dB

7

Weight

120 g

8

Power supply

USB

Description

№

Parameter

Description

1

Capsule type

Electret

2

Frequency response

20 Hz–20 kHz

3

Resolution and sample rate

32; 48; 64 kHz

4

Amplification

≥90 dB

5

Weight

25 g

6

Power supply

Built-in battery

These voice recorders are small in size and have a built-in power system suitable for measurements on high voltage electrical installations. Thus, galvanic isolation is performed, the probability of building potential from a high-voltage installation is reduced. Therefore, such voice recorders can be brought close to a place where a corona discharge occurs. The above gradients of the EF and MF will not lead to the destruction of the electrical circuits of these devices.

3.1 Acoustic Measurements and Their Results The experiment was carried out as follows: the voltage at the insulator gradually increased until a corona discharge occurred and acoustic noise was recorded. The step of increasing the voltage in the presence of a corona discharge was chosen, and such step was held for more than 10 s to obtain a smooth record. The result of the recording fragment is shown in Fig. 1. Only gaps without bursts were separated for further processing. Each of these fragments of the audio file was deployed to a period of 50 Hz. Voice recorders are equipped with an automatic recording level adjustment system. Therefore, the distance from the microphone to the corona discharge cannot be calculated directly from the power of the recorded signal. And each received signal

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Fig. 1 Amplitude values of the acoustic noise of the corona from different voltages on the corona

Fig. 2 Fragments of audio signals deployed to the same period, the amplitudes are increased to a conventional unit (a 31 kV; b 47 kV; c 59 kV; d 71 kV)

was amplified to a conventional unit. The signals were aligned with the maximum amplitude (Fig. 2). According to the shape of the envelope of the curve of the audio signal, there is a difference in shape from the voltage of the corona discharge.

3.2 Spectral Decomposition of Audio Corona Noise Fragments The experimental results were processed in accordance with the rules of mathematical statistics and applied analysis [46–51]. Each of the audio fragments was analyzed for harmonic components. For this, Fourier expansion was used. Fast Fourier transform is used to obtain analysis over the entire frequency range (Fig. 3).

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Fig. 3 Spectral decomposition of audio fragments in accordance with the voltage of the corona discharge (a 31 kV; b 47 kV; c 59 kV; d 71 kV)

The first simplified analysis was performed in 1024 fragments. For frequencies above 4 kHz, the shape of the spectral curve of the audio fragments is almost the same for different corona discharge voltages (Fig. 4). The frequency range from 0 to 4000 Hz has large differences depending on the voltage of the corona discharge (Fig. 5). Difficulties may arise for use in determining corona power over a specified range. First, to determine the amplitude data, it must have a reference signal. And if it is impossible to directly determine the distance to the signal, this method is considered very difficult. Secondly, it is also difficult to determine from the spectrogram that the signal belongs to a corona discharge. Therefore, a more accurate analysis was performed on the Fourier decomposition. The decomposition of the sample with a frequency of 65,536 Hz was used. Thus, in this example, time resolution is 1024/65,536 = 15.6 ms, and the frequency resolution is 64 Hz.

Fig. 4 Spectral curves for various corona discharge voltages for the frequency range 4–20 kHz

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Fig. 5 Spectral curves for various corona discharge voltages for the frequency range 0–4000 Hz

The analysis was carried out in the frequency range from 0 to 500 Hz (Fig. 6). The bursts corresponding to the harmonious components of the industrial frequency are clearly visible. The spectrogram shows even and odd frequencies, which creates a corona discharge in the audio range. Frequency bursts have a narrow range (sharp), therefore, in a simplified analysis, the discreteness stepped over the bursts and could not fix them. Consider separately the harmonic components. 50 Hz (first harmonic)—industrial frequency of the electrical network. Equipment operates at this frequency, which is connected to the network. Acoustic bursts at this frequency can be created not only by corona discharge. Although this harmonic has the greatest values in amplitude, we will not rely on it for now.

Fig. 6 Spectral analysis in the frequency range 0–500 Hz

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100 Hz (second harmonic). For a frequency of 100 ± 1 Hz, it is indicative of an increase of the intensity values by 12 dB compared to the background values. This value can reach −64 dB, that is, such an increase is sufficient for recognition (Fig. 7). 150 Hz (third harmonic). At a frequency of 150 ± 1 Hz, there is an increase of the intensity value by 40 dB compared to the background. This value can reach −29 dB. Such a gradient of intensity values is significant with using this frequency to recognize the presence of a corona discharge (Fig. 8).

Fig. 7 100 Hz frequency for various corona discharge voltages

Fig. 8 150 Hz frequency for various corona discharge voltages

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200 Hz (fourth harmonic). At a frequency of 200 ± 1 Hz, there is an increase of the intensity value by 24 dB compared to the background. And it can reach −51 dB (Fig. 9). 250 Hz (fifth harmonic). At a frequency of 250 ± 1 Hz, there is an increase of the intensity value by 20 dB compared to the background. And it can reach −58 dB (Fig. 10).

Fig. 9 200 Hz frequency for various corona discharge voltages

Fig. 10 250 Hz frequency for various corona discharge voltages

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300 Hz (sixth harmonic). At a frequency of 300 ± 1 Hz, there is an increase of the intensity value by 12 dB compared to the background. And it can reach −66 dB (Fig. 11). 350 Hz (seventh harmonic). At a frequency of 350 ± 1 Hz, there is an increase of the intensity value by 20 dB compared to the background. And it can reach −64 dB (Fig. 12). Further it will be brought all the harmonics values into one table (Table 3).

Fig. 11 300 Hz frequency for various corona discharge voltages

Fig. 12 350 Hz frequency for various corona discharge voltages

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Table 3 Intensity of the corona acoustic depending on the voltage at harmonics (for special conditions) Frequency

31 kV

47 kV

59 kV

71 kV

1 (50 Hz), dB

−1.07

−2.86

−2.25

−3.08

2 (100 Hz), dB

−38.52

−44.77

−45.56

−44.17

3 (150 Hz), dB

−20.73

−21.08

−19.34

−19.82

4 (200 Hz), dB

−34.91

−40.37

−40.57

−45.86

5 (250 Hz), dB

−38.25

−44.40

−44.38

−44.62

6 (300 Hz), dB

−41.88

−48.23

−46.09

−51.10

7 (350 Hz), dB

−51.94

−62.36

−61.43

−54.33

From Table 3 it is clearly seen that the corona discharge is characterized by acoustic bursts at certain frequencies. Odd harmonics have large intensities and are inherent to the corona discharge. Next, it can be traced the dependence. A corona discharge on conductive parts, given an industrial frequency voltage of 50 Hz, causes acoustic disturbance, the spectral picture of which necessarily contains harmonious parts of the following frequencies: 50, 150, 250, 350 Hz and an additional 100, 200, 300 Hz. Based on these dependencies, a model of spectral-acoustic corona discharge parameters was constructed.

4 Recognition of a Corona Discharge in the Presence of Spectral Components In the previous section, the property of certain acoustic spectral components of a corona discharge was shown. Accordingly, the need arose to create a recognition algorithm, that is, to recognize a corona discharge by spectral acoustic components in an acoustic noise stream. The above algorithm is implemented as follows. After starting the process, the microphone picks up acoustic signals and transmits them to the block of analog signals in the form of electrical pulses. The analog signal is processed for the absence of excesses in the amplitudes and is transmitted to the analog-to-digital processing unit. In the analog-to-digital block, the signal is digitized and transmitted for recording in the memory block. Acoustic signal recording is performed continuously during the entire operating time of the analyzing device. From the memory block, the signal is sampled into a block for dividing the file into short fragments (for example, 10 s long). In the next block, the short file is decomposed using the Fourier transform [52–55], where the spectral frequency components of the acoustic signal are extracted. The result of the Fourier transform is cleaned of noise and analyzed for the presence of the frequency spectrum (primary 50–150–250 Hz) and secondary (100–200–300 Hz); when the excess of values coincides for the selected

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signal frequencies over the background, a corresponding synchronization marker with a timer is formed. A corona discharge data stream is generated, which is sent to an external database. Thus, the information flow of fixing the presence of a corona discharge is formed. The data analysis module was created in MatLab. The results of processing acoustic files were obtained at the laboratory bench. Figures 13, 14, 15 and 16 show the following: The upper left graph is the envelope of the acoustic signal received after the ADC. The lower left is a signal stretched to one period (50 Hz). The upper right is the result of the expansion according to the Fourier transform. Bottom right—the spectral lines are increased and the 50 Hz line is excluded (Fig. 17). All of these graphs meet the criteria for the presence of a corona discharge. This confirms the possibility of determining the presence of a corona discharge from the acoustic spectrum.

Fig. 13 Spectral analysis of a fragment of an acoustic file (corona discharge, voltage 31 kV)

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Fig. 14 Spectral analysis of a fragment of an acoustic file (corona discharge, voltage 47 kV)

5 Acoustic Corona Discharge Field Construction To construct the acoustic field, it was considered the equation for the pressure wave [56, 57]. It was wrote the wave equation for three dimensions. Part of the air at the point (x, y, z) can move in an arbitrary direction, so it will write down the three components of its displacement from the equilibrium position. This displacement d is the vector of the components 3 ∂di ξ, η and ε in the x, y, and z directions. The corresponding part velocity has components x, y and z. All these quantities together with pressure u = i=1 ∂t p are functions of x, y, z and t. The elementary volume dxdydz, shifting during the passage of a acoustic wave, turns into a parallelepiped of volume J (dx·dy·dz), where   1 + ∂ξ  ∂x  J =  ∂∂ηx  ∂ς  ∂x

The value div(d) =

∂ξ ∂x

∂ξ ∂y ∂η ∂y ∂ς ∂y

∂ξ ∂z ∂η ∂z

1+

∂ς ∂z

   ∼  = 1 + div(d).  

+ ∂∂ηx + ∂ς is called the particle displacement divergence. ∂x

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Fig. 15 Spectral analysis of a fragment of an acoustic file (corona discharge, voltage 59 kV)

According to the continuity equation: δ=−

1 ∂ξ ;1 + δ = ;δ ∼ = −div(d). ∂x J

According to the equation p = γC P0 δ, where γ C is the ratio of specific heat capacities at constant pressure and constant volume (for air, under normal conditions, takes the value 1.4). For adiabatic gas compression, the relation is takes place: 1+

1 P = (1 + δ)γC = γ ; p ∼ = −γC P0 div(d), P0 J C

where γC = CCPv —ratio of specific heat at constant pressure and constant volume. Newton’s second law gives:

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Fig. 16 Spectral analysis of a fragment of an acoustic file (corona discharge, voltage 71 kV)

p

∂u = −grad( p), ∂t

where grad( p)—vector, which is called the pressure gradient. Its components are equal: ∂∂ px + ∂∂ py + ∂∂zp . These equations can be combined as follows:   γC P0 ∂2 p ∂u = div(grad( p)) = c2 ∇ 2 p, = −γC P0 div ∂t 2 ∂t ρ where c2 =

γC P0 —function ρ

(1)

of pressure, which is determined by the expression

div(grad( p)) =

∂2 p ∂2 p ∂2 p + 2 + 2. ∂x2 ∂y ∂z

This expression is called Laplacian. Formula (1) is a three-dimensional wave equation (for pressure). The particle velocity for a simple harmonic wave can be obtained from pressure using the dependence

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Fig. 17 Stem histogram of the image of the frequency spectrum of the corona discharge at voltage: a 31 kV; b 47 kV; c 59 kV; d 71 kV

u=

1 grad( p). r ωρ

(2)

The potential energy of the gas volume in some instantaneous state can be expressed using the absolute temperature: ρC0 T =

P0 C0 P0 (1 + δ)γC P0 · = . (1 + δ)γC = R (1 + δ) (γC − 1) (γC − 1)J γC −1

0 The equilibrium value (γCP−1) is the internal energy of the gas at equilibrium and is not included in the expression for the energy of a acoustic wave. If it was expanded the expression J γC1 −1 in a series in small quantities ∂∂ξx , etc., then these terms containing these quantities in the first degree, as well as their multiplies of the form ∂ς , ∂η , etc., ∂ y ∂z cannot be ignored, since the average value of these terms is zero. The mean values , etc., did not go to zero: of only those terms that contain the squares of ∂η ∂y

P0 (γC − 1)



1 J γC −1

 −1 ∼ =

P0 , (γC − 1)

C because (γC −1)γ div 2 d—neglected members. 2 The total average energy of the acoustic wave is equal to the integral [58]:

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˚ 

∂d ∂t



2

+ c div d d xd ydz = 2

2

˚ 

1 2 1 2 d xd ydz. ρu + p 2 2ρc2

It was introduced the initial data for a point acoustic emitter—corona discharge. According to measurements made in [59, 60], the average power loss per corona discharge is from 129.5 to 83.3 kW/km (for lines with a voltage of 500 kV). Let’s take the average value of 100 kW/km or 100 W/m. The overall dimensions of the acoustic emitter constitute a sphere with a radius of 0.2 m. The source of acoustic waves from the corona discharge will be considered as a point one. In accordance with the theory of acoustics, a “point” emitter, that is, an emitter whose dimensions can be neglected in comparison with the length of the acoustic wave emitted by it.

5.1 Construction of a Curve of Acoustic Strength from Corona Discharge Elements The direction of radiation of acoustic waves in the theory of acoustics is called the radiation pattern. It was expanded this term and assumed that the radiation pattern is a combination of the extremities of power vectors (acoustic pressure forces) in space. For a flat view, sections of the volumetric body are used with planes—a vertical plane and a horizontal plane. Then it can be taught about the curve of acoustic power (CAP). For corona discharge at the insulator, CAP has the form shown in Figs. 18 and 19. Fig. 18 Acoustic corona discharge curve on insulator

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Fig. 19 CAP from the corona in the insulator in space

At the same time, for the corona discharge on the wire, the CAP has the form presented in Fig. 20. The occurrence of a corona discharge at one point does not lead to significant losses on the line and, accordingly, does not lead to significant acoustic emissions. But a corona discharge occurs simultaneously in several places of conductive equipment (Fig. 21). Therefore, it can be considered the source of the acoustic signal from the corona discharge as simultaneous in space. It was constructed the CAP as simultaneous from the corona discharge on insulators (Fig. 22) and on the wire (Fig. 23).

Fig. 20 CAP from the corona on the wire

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Fig. 21 Garland of insulators for calculating the CAP

5.2 Calculation of the Acoustic Field from Elements with a Corona Discharge Calculation of the acoustic field from the garland of insulators with corona discharge is carried out according to [61, 62]. To calculate the acoustic field, it was taken a rectangular space in the center of which it will arrange a garland of insulators (Fig. 24), where x—length, y—width, z—height of the space (x = 10 m, y = 10 m, z = 5 m, respectively); hd —height of the research instruments 1.5 m; h—height from a string of insulators to registration devices 3.5 m. It was built a acoustic field isoline picture in the X  Y  plane. It was accepted the number of insulators in the garland N = 5 pcs. It was assumed that a corona will appear on each of them. The acoustic power of each of the acoustic sources is assumed to be W 3 = 1 W. The first calculation is carried out for a frequency of 50 Hz. Calculations are performed for the energy and interference fronts of the acoustic field.

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Fig. 22 CAP from simultaneous corona discharge on several insulators

The picture of the acoustic field isolines for a frequency of 50 Hz shows that the detection of a corona discharge at this frequency is very problematic (Fig. 25). The average gradient is very small. Acoustic field strength is less than 54 dB. But it turned out that for the interference picture of the acoustic field, it is likely to find even the source of the location of the corona discharge, which creates an acoustic peak in the projection of the insulator onto the modeling plane. The gradient of the acoustic field in the interference calculation can be more than 2 dB, which is already noticeable for measuring instruments for finding the coordinates of the acoustic source. The picture of the isolines of the acoustic field for the third harmonic with a frequency of 150 Hz shows that the detection of a corona discharge at this frequency by energy calculation can be performed (Fig. 26). The acoustic field strength is 82– 84 dB, which is sufficient. The average gradient is 6 dB. But the interference picture of the acoustic field has the interference radii of the powerful front 86 dB more than 5 m. The gradient of the acoustic field in the interference calculation is almost 0 dB, which is problematic for finding the acoustic source at such a distance. The picture of the isolines of the acoustic field for the fifth harmonic with a frequency of 250 Hz shows that the detection of a corona discharge at this frequency by energy calculation can be performed. The acoustic field strength is 78–80 dB,

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Fig. 23 CAP from simultaneous corona discharge in several places on the wire

Fig. 24 Space for calculating the acoustic field from a corona source (a string of insulators)

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Fig. 25 Result of the calculation of the acoustic field isolines for a garland of 5 insulators at a frequency of 50 Hz (on the left are the energy fronts of the acoustic field, on the right are the interference fronts)

Fig. 26 Result of the calculation of the acoustic field isolines for a garland of 5 insulators at a frequency of 150 Hz (on the left are the energy fronts of the acoustic field, on the right are the interference fronts)

which is sufficient. The average gradient is 6 dB. But the interference pattern of the acoustic field has interference radii with a peak in projection to the acoustic source of 70 dB, and a significant gradient of the acoustic field, which is almost 10 dB, is sufficient enough to find the acoustic source (Fig. 27).

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Fig. 27 Result of the calculation of the acoustic field isolines for a garland of 5 insulators at a frequency of 250 Hz (on the left are the energy fronts of the acoustic field, on the right are the interference fronts)

5.3 Calculation of the Acoustic Field from a Corona Wire To calculate the acoustic field, it was taken a rectangular space in the center of which a wire with an acoustic signal source arranged, which should simulate a corona discharge acoustic source (Fig. 28), where x—length, y—width, z—height of the

Fig. 28 Space for calculating the acoustic field from a corona source (cable-wire)

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Fig. 29 Result of the calculation of the acoustic field isolines for a wire with 10 sources at a frequency of 50 Hz (on the left are the energy fronts of the acoustic field, on the right are the interference fronts)

space (x = 20 m, y = 20 m, z = 10 m, respectively); hd —height of the research instruments 1.5 m; h—height from a string of insulators to registration devices 8.5 m. It was accepted the number of insulators in the garland N = 10 pcs. The acoustic power of each of the acoustic sources is assumed to be W 3 = 1 W. The first calculation is carried out for a frequency of 50 Hz. Calculations are performed for the energy and interference fronts of the acoustic field (Fig. 29). The picture of the acoustic field isolines for a frequency of 50 Hz shows that the detection of a corona discharge at this frequency is very problematic. The average gradient is very small. Acoustic field strengths are less than 54 dB. But for the interference picture of the acoustic field, it is likely to find the source of the location of the corona discharge, which creates an acoustic peak not only in the projection of the wire onto the modelling plane, but also the maximum of the gradient is stretched to a distance of 5 m. The acoustic field strength at the epicenter of the interference picture is 62–64 dB. The gradient of the acoustic field in the interference calculation can be more than 10 dB, which is noticeable for measuring instruments to search for the coordinates of the acoustic source. The picture of the acoustic field isolines for the third harmonic frequency of 150 Hz shows that the detection of a corona discharge at this frequency according to energy calculations can be performed (Fig. 30). The acoustic field strength is 82–84 dB, which is very sufficient. The average gradient is 6 dB. But the interference picture of the acoustic field has a powerful front radius of 94–96 dB with an epicenter of more than 5 m. The gradient of the acoustic field in the interference calculation is almost 20 dB in the direction perpendicular to the wire and at distances greater than 10 m. This is more than sufficient to detect a acoustic source. The picture of the isolines of the acoustic field for the fifth harmonic frequency of 250 Hz shows that the detection of a corona discharge at this frequency by energy calculation can be performed (Fig. 31). The acoustic field strength is 78–80 dB, which

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Fig. 30 Result of the calculation of the acoustic field isolines for a wire with 10 sources at a frequency of 150 Hz (on the left are the energy fronts of the acoustic field, on the right are the interference fronts)

Fig. 31 Result of the calculation of the acoustic field isolines for a wire with 10 sources at a frequency of 250 Hz (on the left are the energy fronts of the acoustic field, on the right are the interference fronts)

is sufficient. The average gradient is 8 dB at the estimated distance. In addition, the interference pattern of the acoustic field has a significantly pronounced peak directed perpendicular to the acoustic source and with an acoustic intensity at the epicenter of 84–86 dB, and a significant gradient of the acoustic field, which is more than 20 dB, allows to find the acoustic source. From isoline maps for various acoustic frequencies (50, 150, 250 Hz) and various types of sources, it is clearly seen that there are areas where corona discharge detection is most likely. Such zones should be used for devices having a significant intrinsic speed, which will allow them to determine the coordinates of the acoustic source,

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that is, the coordinates of the corona discharge. It also allowed to perform quality control of the electrical network [63–68].

References 1. Gryb, O., Karpaliuk, I., Shvets, S., Zaporozhets, A.: Recognition of corona discharge presence by acoustic system installed on unmanned aerial vehicle. In: Proceedings of the National Aviation University, vol. 85, pp. 46–53 (2020). https://doi.org/10.18372/2306-1472.85.15138 2. Abahazem, A., Merbahi, N., Guedah, H., Yousfi, M.: Electric and spectroscopic studies of pulsed corona discharges in nitrogen at atmospheric pressure. J. Anal. Sci. Methods Instrum. 7(3), 77870 (2017). https://doi.org/10.1088/0963-0252/25/4/045003 3. Siva Sarma, D.V.S.S., Kalyani, G.N.S.: ANN approach for condition monitoring of power transformers using DGA. In: 2004 IEEE Region 10 Conference TENCON 2004, 24 Nov. 2004, Chiang Mai, Thailand, pp. 444–447 (2004). https://doi.org/10.1109/TENCON.2004.141 4803 4. Ashmarin, G.V., Lelevkin, V., Niyazaliev, I.A., Tokarev, A.V.: The estimation of steel rope quality by a corona discharge. In: 5-International Conference Plasma Physics and Plasma Technologies, Minsk, Belarus, vol. 2, pp. 808–811 (2006) 5. Niedbalski, J.: High-voltage multichannel rail gap switch triggered by corona discharges. Rev. Sci. Instrum. 74, 3520 (2003). https://doi.org/10.1063/1.1584081 6. Pucharev, V., Gundersen, M.: Energy efficient plasma processing of gaseous emission using a short pulse discharge. Appl. Phys. Lett. 71(23), 3364 (1997). https://doi.org/10.1063/1.120338 7. Burany, N., Huber, L., Pejovic, P.: Corona discharge surface treater without high voltage transformer. IEEE Trans. Power Electron. 23(2), 993–1002 (2008). https://doi.org/10.1109/TPEL. 2007.915760 8. Mok, Y.S., Nam, I.-S.: Modeling of pulsed corona discharge process for the removal of nitric oxide and sulfur dioxide. Chem. Eng. J. 85(1), 87–97 (2002). https://doi.org/10.1016/S13858947(01)00221-2 9. Filatov, I.E., Uvarin, V.V., Kuznetsov, D.L.: Cleaning air from multicomponent impurities of volatile organic compounds by pulsed corona discharge. Tech. Phys. Lett. 42, 927–931 (2016). https://doi.org/10.1134/S1063785016090169 10. Filatov, I.E., Uvarin, V.V., Kuznetsov, D.L.: Estimation of qualitative and quantitative parameters of air cleaning by a pulsed corona discharge using multicomponent standard mixtures. Tech. Phys. 63, 680–688 (2018). https://doi.org/10.1134/S1063784218050079 11. Rahimi, M.R., Javadinezhad, R., Vakilian, M.: DC partial discharge characteristics for corona, surface and void discharges. In: 2015 IEEE 11th International Conference on the Properties and Applications of Dielectric Materials (ICPADM), 19–22 July 2015, Sydney, NSW, Australia, pp. 260–263 (2015) https://doi.org/10.1109/ICPADM.2015.7295258 12. Pileggi, D.J., Harish Chandra, N., Emanuel, A.E.: Prediction of harmonic voltages in distribution systems. IEEE Trans. Power Appar. Syst. PAS-100, 3, pp. 1307–1315 (1981) https://doi. org/10.1109/TPAS.1981.316603 13. Bashir, N., Ahmad, H.: Odd harmonics and third to fifth harmonic ratios of leakage currents as diagnostic tools to study the ageing of glass insulators. IEEE Trans. Dielectr. Electr. Insul. 17(3), 819–832 (2010). https://doi.org/10.1109/TDEI.2010.5492255 14. Samimi, M.H., Mostajabl, A.H., Ahmadi-Joneidi, I., Shayegani-Akmal, A.A., Mohseni, H.: Performance evaluation of insulators using flashover voltage and leakage current. Electr. Power Compon. Syst. 41(2), 221–233 (2013). https://doi.org/10.1080/15325008.2012.738354 15. Ajao, K.R., Ajimotokan, H.A., Popoola, O.T., Akande, H.F.: Electric energy supply in Nigeria, decentralized energy approach. Cogener. Distrib. Gener. J. 24(4), 34–50 (2009). https://doi. org/10.1080/15453660909595149

154

O. G. Gryb et al.

16. Liu, Y.-J., Chang, T.-P., Chen, H.-W., Chang, T.-K., Lan, P.-H.: Power quality measurements of low-voltage distribution system with smart electric vehicle charging infrastructures. In: 2014 16th International Conference on Harmonics and Quality of Power (ICHQP), 25–28 May 2014, Bucharest, Romania, pp. 631–635. https://doi.org/10.1109/ICHQP.2014.6842879 17. Kordkheilli, H.H., Abravesh, H., Tabasi, M., Dakhem, M., Abravesh, M.M.: Determining the probability of flashover occurrence in composite insulators by using leakage current harmonic components. IEEE Trans. Dielectr. Electr. Insul. 17(2), 502–512 (2010). https://doi.org/10. 1109/TDEI.2010.5448106 18. Catterson, V.M., Bahadoorsingh, S., Rudd, S., McArthur, S.D.J., Rowland, S.M.: Identifying harmonic attributes from online partial discharge data. IEEE Trans. Power Deliv. 26(3), 1811– 1819 (2011). https://doi.org/10.1109/TPWRD.2011.2114373 19. Tungkanawanich, A., Kawasaki, Z.-I., Abe, J., Matsuura, K.: Location of partial discharge source on distribution line by measuring emitted pulse-train electromagnetic waves. In: 2000 IEEE Power Engineering Society Winter Meeting. Conference Proceedings (Cat. No.00CH37077), 23–27 Jan. 2000, Singapore, pp. 2453–2458. https://doi.org/10.1109/PESW. 2000.847194 20. Alvarez, F., Garnacho, F., Ortego, J., Sanchez-Uran, M.A.: Application of HFCT and UHF sensors in on-line partial discharge measurements for insulation diagnosis of high voltage equipment. Sensors 15(4), 7360–7387 (2015). https://doi.org/10.3390/s150407360 21. Chen, L., MacAlpine, J.M.K., Bian, X., Wang, L., Guan, Z.: Comparison of methods for determining corona inception voltages of transmission line conductors. J. Electrostat. 71(3), 269–275 (2013). https://doi.org/10.1016/j.elstat.2012.11.020 22. Kachi, M., Nemamcha, M., Lazhar, H., Dascalescu, L.: Neutralization of charged insulating granular materials using AC corona discharge. J. Electrostat. 69(4), 296–301 (2011). https:// doi.org/10.1016/j.elstat.2011.04.005 23. Adamiak, K., Atten, P.: Simulation of corona discharge in point–plane configuration. J. Electrostat. 61(2), 85–98 (2004). https://doi.org/10.1016/j.elstat.2004.01.021 24. Wallis, J.: Making the invisible visible: UViRCO, an innovation success story: industry. CSIR Sci. Scope 8(2), 80–81 (2015) 25. Karady, G.G., Besztercey, G., Tuominen, M.W.: Corona caused deterioration of ADSS fiberoptic cables on high voltage lines. IEEE Trans. Power Deliv. 14(4), 1438–1447 (1999). https:// doi.org/10.1109/61.796238 26. Shong, K.-M., Kim, Y.-S., Kim, S.-G.: Images detection and diagnosis of corona discharge on porcelain insulators at 22.9 kV D/L. In: 2007 IEEE International Symposium on Diagnostics for Electric Machines, Power Electronics and Drives, 6–8 July 2007, Cracow, Poland, pp. 462–466. https://doi.org/10.1109/DEMPED.2007.4393138 27. Chen, J., Davidson, J.H.: Ozone production in the positive DC corona discharge: model and comparison to experiments. Plasma Chem. Plasma Process. 22, 495–522 (2002). https://doi. org/10.1023/A:1021315412208 28. Van Durme, J., Dewulf, J., Sysmans, W., Leys, C., Van Langenhove, H.: Abatement and degradation pathways of toluene in indoor air by positive corona discharge. Chemosphere 68(10), 1821–1829 (2007). https://doi.org/10.1016/j.chemosphere.2007.03.053 29. Lukes, P., Clupek, M., Babicky, V., Janda, V., Sunka, P.: Generation of ozone by pulsed corona discharge over water surface in hybrid gas–liquid electrical discharge reactor. J. Phys. D Appl. Phys. 38(3), 409 (2005). https://doi.org/10.1088/0022-3727/38/3/010 30. Novak, I., Pollak, V., Chodak, I.: Study of surface properties of polyolefins modified by corona discharge plasma. Plasma Process. Polym. 3(4–5), 355–364 (2006). https://doi.org/10.1002/ ppap.200500163 31. Urakseev, M.A., Vazhdaev, K.V., Sagadeev, A.R.: Fiber-optical sensor with an acousto-optical filter for monitoring the status of overhead power lines. In: 2019 International Ural Conference on Electrical Power Engineering (UralCon), 1–3 Oct. 2019, Chelyabinsk, Russia, pp. 97–101. https://doi.org/10.1109/uralcon.2019.8877632 32. Li, Q., Shuttleworth, R., Zhang, G., Dupere, I., Rowland, S.M.: Acoustic noise evaluation for overhead line conductors. In: 2013 IEEE Electrical Insulation Conference (EIC), 2–5 June 2013, Ottawa, ON, Canada, pp. 119–123. https://doi.org/10.1109/EIC.2013.6554216

Acoustic Diagnostics for Determining the Appearance …

155

33. Suljanovic, N., Mujcic, A., Zajc, M., Tasic, J.F.: Corona noise characteristics in high voltage PLC channel. In: IEEE International Conference on Industrial Technology, 2003, 10–12 Dec. 2003, Maribor, Slovenia, pp. 1036–1039. https://doi.org/10.1109/ICIT.2003.1290805 34. Suljanovic, N., Mujcic, A., Zajc, M., Tasic, J.F.: Computation of high-frequency and time characteristics of corona noise on HV power line. IEEE Trans. Power Deliv. 20(1), 71–79 (2005). https://doi.org/10.1109/TPWRD.2004.838656 35. Otto, A.J., Reader, H.C.: Wideband and narrowband HVDC conductor corona test methods for radio noise prediction. IEEE Trans. Power Delivery 25(4), 2950–2957 (2010). https://doi.org/ 10.1109/TPWRD.2010.2051689 36. Stone, G.C.: Partial discharge diagnostics and electrical equipment insulation condition assessment. IEEE Trans. Dielectr. Electr. Insul. 12(5), 891–904 (2005). https://doi.org/10.1109/tdei. 2005.1522184 37. Glowacz, A., Glowacz, W., Glowacz, Z., Kozik, J.: Early fault diagnosis of bearing and stator faults of the single-phase induction motor using acoustic signals. Measurement 113, 1–9 (2018). https://doi.org/10.1016/j.measurement.2017.08.036 38. Abramov, I.V., Abramov, A.I., Nikitin, Y.R., Sosnovich, E., Bozek, P., Stollmann, V.: Diagnostics of electrical drives. In: 2015 International Conference on Electrical Drives and Power Electronics (EDPE), 21–23 Sept. 2015, Tatranska Lomnica, Slovakia, pp. 364–367 (2015). https://doi.org/10.1109/edpe.2015.7325321 39. Su, C.-C., Tai, C.-C., Chen, C.-Y., Hsieh, J.-C., Chen, J.-F.: Partial discharge detection using acoustic emission method for a waveguide functional high-voltage cast-resin dry-type transformer. In: 2008 International Conference on Condition Monitoring and Diagnosis, 21–24 April 2008, Beijing, China, pp. 517–520. https://doi.org/10.1109/cmd.2008.4580339 40. Akumu, A.O., Adachi, F., Kawaguchi, N., Ozaki, R., Ihori, H., Fujii, M., Arii, K.: A 3-D numerical simulation of partial discharge acoustic wave propagation in a model transformer. In: Conference Record of the 2002 IEEE International Symposium on Electrical Insulation (Cat. No. 02CH37316), 7–10 April 2002, Boston, MA, USA, pp. 183–186. http://doi.org/10. 1109/ELINSL.2002.995908 41. Beguin, Ph., Joly, V., Herzog, Ph: Modeling of a corona discharge microphone. J. Phys. D Appl. Phys. 46(47), (2013). https://doi.org/10.1088/0022-3727/46/17/175204 42. Kopiev, V.F., Zaitsev, MYu., Kopiev, V.A., Ostrikov, N.N., Faranosov, G.A.: Application of corona discharge acoustic characteristics to determine its properties. Acoust. Phys. 62, 429–435 (2016). https://doi.org/10.1134/S1063771016040084 43. Leger, L., Moreau, E., Touchard, G.G.: Effect of a DC corona electrical discharge on the airflow along a flat plate. IEEE Trans. Ind. Appl. 38(6), 1478–1485 (2002). https://doi.org/10.1109/ tia.2002.804769 44. Cagnoni, D., Agostini, F., Christen, T., Parolini, N., Stevanovic, I., de Falco, C.: Multiphysics simulation of corona discharge induced ionic wind. J. Appl. Phys. 114(23), (2013). https://doi. org/10.1063/1.4843823 45. Khalifa, M., Abdel-Salam, M.: The corona discharge. Electr. Comput. Eng. 149–184 (2000) 46. Ihlenburg, F.: Finite Element Analysis of Acoustic Scattering, vol. 132. Springer Science & Business Media (2006) 47. Durand, J.-F., Soize, C.: Structural-acoustic modeling of automotive vehicles in presence of uncertainties and experimental identification and validation. J. Acoust. Soc. Am. 124, 1513 (2008). https://doi.org/10.1121/1.2953316 48. Tanner, G., Sondergaard, N.: Wave chaos in acoustics and elasticity. J. Phys. A: Math. Theor. 40(50), R443 (2007). https://doi.org/10.1088/1751-8113/40/50/R01 49. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Methods and models for information data analysis. In: Diagnostic Systems for Energy Equipments. Studies in Systems, Decision and Control, vol. 281, pp. 23–70 (2020). https://doi.org/10.1007/978-3030-44443-3_2 50. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Simulation and software for diagnostic systems. In: Diagnostic Systems For Energy Equipments. Studies in Systems, Decision and Control, vol. 281, pp. 71–90. Springer, Cham (2020). https://doi.org/ 10.1007/978-3-030-44443-3_3

156

O. G. Gryb et al.

51. Babak, V.P., Babak, S.V., Myslovych, M.V., Zaporozhets, A.O., Zvaritch, V.M.: Technical provision of diagnostic systems. In: Diagnostic Systems For Energy Equipments. Studies in Systems, Decision and Control, vol. 281, pp. 91–133 (2020). https://doi.org/10.1007/978-3030-44443-3_4 52. Babak, V., Eremenko, V., Zaporozhets, A.: Research of diagnostics parameters of composite materials using Johnson distribution. Int. J. Comput. 18(4), 483–494 (2019) 53. Zaporozhets, A., Babak, V., Sverdlova, A., Isaienko, V., Babikova, K.: Development of a system for diagnosing heat power equipment based on IEEE 802.11s. In: Zaporozhets, A., Artemchuk, V. (eds) Systems, Decision and Control in Energy II. Studies in Systems, Decision and Control, pp. 141–151. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-69189-9_8 54. Zaporozhets, A., Eremenko, V., Serhiienko, R., Ivanov, S.: Methods and hardware for diagnosing thermal power equipment based on smart grid technology. In: Shakhovska, N., Medykovskyy, M. (eds.) Advances in Intelligent Systems and Computing III. CSIT 2018. Advances in Intelligent Systems and Computing, vol. 871, pp. 476–489. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-01069-0_34 55. Eremenko, V., Zaporozhets, A., Isaenko, V., Babikova, K.: Application of wavelet transform for determining diagnostic signs. In: CEUR Workshop Proceedings, vol. 2387, pp. 202–214. http://ceur-ws.org/Vol-2387/20190202.pdf 56. Jenkins, E.W.: Numerical solution of the acoustic wave equation using Raviart-Thomas elements. J. Comput. Appl. Math. 206(1), 420–431 (2007). https://doi.org/10.1016/j.cam.2006. 08.003 57. Ehrendorfer, K., Reiterer, M., Sockel, H.: Numerical investigation of the micro pressure wave. In: Schulte-Werning, B., Grégoire, R., Malfatti, A., Matschke, G. (eds.) TRANSAERO—A European Initiative on Transient Aerodynamics for Railway System Optimisation. Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), vol. 79, pp. 321–341. Springer, Berlin, Heidelberg (2002). https://doi.org/10.1007/978-3-540-45854-8_26 58. Mkrtchyan, A.R., Hayrapetyan, A.G., Khachatryan, B.V., Petrosyan, R.G.: Transformation of sound and electromagnetic waves in non-stationary media. Mod. Phys. Lett. B 24(18), 1951–1961 (2010). https://doi.org/10.1142/S021798491002433X 59. Sobacchi, M.G., Saveliev, A.V., Fridman, A.A., Gutsol, A.F., Kennedy, L.A.: Experimental assessment of pulsed corona discharge for treatment of VOC emissions. Plasma Chem. Plasma Process. 23, 347–370 (2003). https://doi.org/10.1023/A:1022976204132 60. Al-Hamouz, Z., El-Hamouz, A., Abuzaid, N.: Simulation and experimental studies of corona power loss in a dust loaded wire-duct electrostatic precipitator. Adv. Powder Technol. 22(6), 706–714 (2011). https://doi.org/10.1016/j.apt.2010.10.005 61. Lüthi, B.: Physical Acoustics in the Solid State, vol. 148. Springer Science & Business Media (2007) 62. Ha, H., Han, S., Lee, J.: Fault detection on transmission lines using a microphone array and an infrared thermal imaging camera. IEEE Trans. Instrum. Meas. 61(1), 267–275 (2012). https:// doi.org/10.1109/TIM.2011.2159322 63. Babak, S., Babak, V., Zaporozhets, A., Sverdlova, A.: Method of statistical spline functions for solving problems of data approximation and prediction of objects state. In: CEUR Workshop Proceedings, vol. 2353, pp. 810–821 (2019). http://ceur-ws.org/Vol-2353/paper64.pdf 64. Artemchuk, V.O., et al.: Theoretical and applied bases of economic, ecological and technological functioning of energy objects (2017) 65. Babak, V.P.: Hardware-software for monitoring the objects of generation, transportation and consumption of thermal energy (2016). ISBN 978-966-02-7967-4 66. Zaporozhets, A., Eremenko, V., Isaenko, V., Babikova, K.: Approach for creating reference signals for detecting defects in diagnosing of composite materials. In: Shakhovska N., Medykovskyy M. (eds.) Advances in Intelligent Systems and Computing IV. CCSIT 2019. Advances in Intelligent Systems and Computing, vol. 1080, pp. 154–172, Springer, Cham (2020). https://doi.org/10.1007/978-3-030-33695-0_12

Acoustic Diagnostics for Determining the Appearance …

157

67. Fuchs, E., Masoum, M.A.: Power Quality in Power Systems and Electrical Machines. Academic Press (2011) 68. Sankaran, C.: Power Quality. CRC Press (2017)