Abstract: This paper proposes a new technique for analyzing the recovery voltages in circuit breakers of high voltage networks comprising strongly non-uniform lines. As a representative example, the model considers transmission lines comprising multiple cascade-connected tower spans the resulting information is crucial for assessing the proper operation and protection of those networks. The mathematical model is based on a system composed of simultaneous partial differential and algebraic equations. They formulate the interruption of fault currents at their zero crossings. They can be solved numerically in terms of parametric functions using the software Mathematica using the statement (ParametricNDSolve). The resulting currents and voltage distributions along the line will be functions of the location and the Laplace operator s, as well as of the lines’ loading conditions. The paper includes also a code which is used for the numerical Laplace inversion of these expressions using the well-known Hosono algorithm. The results pertinent to several case studies are presented and discussed in order to demonstrate the impact of the towers’ number and data, as well as the fault current on the breakers’ transient recovery voltages.
It is believed that this study will be helpful to the high voltage and protection engineers in addition to power engineering students and trainees.

Key Words: Electromagnetic, Transients, Simulation, High Voltage Lines, Non-Uniform, Mathematica, Towers, Span, Differential Equations, Parametric Functions, Numerical Laplace Inversion Hosono Algorithm, Parameter Study.

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1. Grainger JJ. Power system analysis. McGraw-Hill; 1999.
2. Bewley LV. Traveling waves on transmission systems. Transactions of the American Institute of Electrical Engineers. 1931 Jun;50(2):532-50.
3. Greenwood A. Electrical transients in power systems. 2nd edition, 01 January 1991, U.S
4. Dommel HW. Electromagnetic Transients Program Reference Manual:(EMTP) Theory Book. Bonneville Power Administration; 1986.
5. Van der Sluis L. Transients in power systems. New York: Wiley; 2001 Aug 8.
6. Begamudre RD. Extra high voltage AC transmission engineering. New Age International; 2006.
7. Martinez-Velasco JA, editor. Transient analysis of power systems: solution techniques, tools and applications. John Wiley & Sons; 2015 Jan 27.
8. Shenkman AL. Transient analysis of electric power circuits handbook. Springer Science & Business Media; 2006 Jan 16.
9. Machowski J, Lubosny Z, Bialek JW, Bumby JR. Power system dynamics: stability and control. John Wiley & Sons; 2020 Jun 8.
10. Short TA. Electric power distribution equipment and systems. CRC press; 2018 Oct 3.
11. Das JC. Transients in electrical systems. McGraw-Hill Professional Publishing; 2010.
12. Rüdenberg R. Transient performance of electric power systems: phenomena in lumped networks. (No Title). 1950.
13. Ovick NL. Travelling waves on transmission lines including corona effects. University of Pittsburgh; 1982.
14. Alvarado FL, Lasseter RH, Won HK. A module oriented EMTP interface. IEEE transactions on power apparatus and systems. 1984 Dec(12):3488-95.
15. SAIED MM, SAFAR YA. Electromagnetic transients on compensated lines under corona. Electric Machines and Power Systems. 1989 Jan 1;16(6):441-62.
16. Ametani A, Nagaoka N, Baba Y, Ohno T, Yamabuki K. Power system transients: theory and applications. CRC Press; 2016 Nov 18.
17. SAIED MM, SAFAR YA. Electromagnetic transients on compensated lines under corona. Electric Machines and Power Systems. 1989 Jan 1;16(6):441-62.
18. Mohamed Mostafa Saied:” Combined Effect of both the Conductors’ Sag and the Initiating Stimuli on the Electromagnetic Transients in High Voltage Lines”, International Journal of Electrical and Communication Engineering Technology (2024)
19. Saied MM. Corona Modeling for the Transient Analysis, Steady State and Corona Loss Performance of Transmission Lines. Electrical Power Quality and Utilisation. Journal. 2013;16.
20. Sofroniou M, Knapp R. Wolfram Mathematica© Tutorial Collection. Advanced Numerical Differential Equation Solving in Mathematica; Wolfram Resarch. Inc.: Champaign, IL, USA. 2008.
21. Hosono T. Numerical inversion of Laplace transforms and some applications to wave optics. Radio Science. 1981 Nov;16(6):1015-9.
22. Abur A, Kim H, Celik MK. Identifying the unknown circuit breaker statuses in power networks. IEEE Transactions on Power Systems. 1995 Nov;10(4):2029-37.

Abstract: This paper proposes a new technique for analyzing the recovery voltages in circuit breakers of high voltage networks comprising strongly non-uniform lines. As a representative example, the model considers transmission lines comprising multiple cascade-connected tower spans the resulting information is crucial for assessing the proper operation and protection of those networks. The mathematical model is based on a system composed of simultaneous partial differential and algebraic equations. They formulate the interruption of fault currents at their zero crossings. They can be solved numerically in terms of parametric functions using the software Mathematica using the statement (ParametricNDSolve). The resulting currents and voltage distributions along the line will be functions of the location and the Laplace operator s, as well as of the lines’ loading conditions. The paper includes also a code which is used for the numerical Laplace inversion of these expressions using the well-known Hosono algorithm. The results pertinent to several case studies are presented and discussed in order to demonstrate the impact of the towers’ number and data, as well as the fault current on the breakers’ transient recovery voltages.
It is believed that this study will be helpful to the high voltage and protection engineers in addition to power engineering students and trainees.

Key Words: Electromagnetic, Transients, Simulation, High Voltage Lines, Non-Uniform, Mathematica, Towers, Span, Differential Equations, Parametric Functions, Numerical Laplace Inversion Hosono Algorithm, Parameter Study.

🔒 This is a subscription article

Full text is available to subscribers and institutional members. Please choose an option below to access it.

1. Grainger JJ. Power system analysis. McGraw-Hill; 1999.
2. Bewley LV. Traveling waves on transmission systems. Transactions of the American Institute of Electrical Engineers. 1931 Jun;50(2):532-50.
3. Greenwood A. Electrical transients in power systems. 2nd edition, 01 January 1991, U.S
4. Dommel HW. Electromagnetic Transients Program Reference Manual:(EMTP) Theory Book. Bonneville Power Administration; 1986.
5. Van der Sluis L. Transients in power systems. New York: Wiley; 2001 Aug 8.
6. Begamudre RD. Extra high voltage AC transmission engineering. New Age International; 2006.
7. Martinez-Velasco JA, editor. Transient analysis of power systems: solution techniques, tools and applications. John Wiley & Sons; 2015 Jan 27.
8. Shenkman AL. Transient analysis of electric power circuits handbook. Springer Science & Business Media; 2006 Jan 16.
9. Machowski J, Lubosny Z, Bialek JW, Bumby JR. Power system dynamics: stability and control. John Wiley & Sons; 2020 Jun 8.
10. Short TA. Electric power distribution equipment and systems. CRC press; 2018 Oct 3.
11. Das JC. Transients in electrical systems. McGraw-Hill Professional Publishing; 2010.
12. Rüdenberg R. Transient performance of electric power systems: phenomena in lumped networks. (No Title). 1950.
13. Ovick NL. Travelling waves on transmission lines including corona effects. University of Pittsburgh; 1982.
14. Alvarado FL, Lasseter RH, Won HK. A module oriented EMTP interface. IEEE transactions on power apparatus and systems. 1984 Dec(12):3488-95.
15. SAIED MM, SAFAR YA. Electromagnetic transients on compensated lines under corona. Electric Machines and Power Systems. 1989 Jan 1;16(6):441-62.
16. Ametani A, Nagaoka N, Baba Y, Ohno T, Yamabuki K. Power system transients: theory and applications. CRC Press; 2016 Nov 18.
17. SAIED MM, SAFAR YA. Electromagnetic transients on compensated lines under corona. Electric Machines and Power Systems. 1989 Jan 1;16(6):441-62.
18. Mohamed Mostafa Saied:” Combined Effect of both the Conductors’ Sag and the Initiating Stimuli on the Electromagnetic Transients in High Voltage Lines”, International Journal of Electrical and Communication Engineering Technology (2024)
19. Saied MM. Corona Modeling for the Transient Analysis, Steady State and Corona Loss Performance of Transmission Lines. Electrical Power Quality and Utilisation. Journal. 2013;16.
20. Sofroniou M, Knapp R. Wolfram Mathematica© Tutorial Collection. Advanced Numerical Differential Equation Solving in Mathematica; Wolfram Resarch. Inc.: Champaign, IL, USA. 2008.
21. Hosono T. Numerical inversion of Laplace transforms and some applications to wave optics. Radio Science. 1981 Nov;16(6):1015-9.
22. Abur A, Kim H, Celik MK. Identifying the unknown circuit breaker statuses in power networks. IEEE Transactions on Power Systems. 1995 Nov;10(4):2029-37.