Comparative Assessment of Numerical Integration of the Equation of Motion of an Electric Train for Automatic Train Operation

Oleg S. Valinskiy; Andrey Evstaf'ev; Anton Sychugov; Evgeniy Suhanov

doi:10.20295/2223-9987-2026-3-7-22

UDC 629.4.053

COMPARATIVE ASSESSMENT OF NUMERICAL INTEGRATION OF THE EQUATION OF MOTION OF AN ELECTRIC TRAIN FOR AUTOMATIC TRAIN OPERATION Comparative Assessment of Numerical Integration of the Equation of Motion of an Electric Train for Automatic Train Operation

АА Аа

Published in Bulletin of scientific research results · Issue 3, 2026 · Pages 7–22 · Rubric: PROBLEMATIC OF TRANSPORT SYSTEM

DOI: https://doi.org/10.20295/2223-9987-2026-3-7-22

Received: 21.06.2026 Accepted: 11.08.2026 Published: 05.10.2026 Language of publication: RUS

Authors

Oleg S. Valinskiy

Andrey Evstaf'ev

Anton Sychugov

Evgeniy Suhanov

Oleg S. Valinskiy

Academic degree: candidate of technical sciences

Science rang: docent

Emperor Alexander I Petersburg State Transport University Russian Federation

Andrey Evstaf'ev

Academic degree: doctor of technical sciences

Science rang: professor

Emperor Alexander I St. Petersburg State Transport University Head of the Department · Department of Electric Traction · Russian Federation

Anton Sychugov

Academic degree: candidate of technical sciences

Emperor Alexander I St. Petersburg State Transport University Associate Professor · Russian Federation · Saint-Petersburg

Evgeniy Suhanov

Emperor Alexander I St. Petersburg State Transport University Postgraduate Student · Russian Federation · St. Petersburg

ABSTRACT

Objective: comparative assessment of variants of numerical integration of the longitudinal equation of motion of an electric train and identification of rational combinations of method and step size for the computational core of an automatic train operation system according to the criteria of accuracy and computation time. Methods: integration of the longitudinal equation of motion with respect to time, speed and distance was considered using separate sets of numerical methods. For integration with respect to distance, an energy transformation of the equation was additionally investigated. For the methods considered, comparative simulation of the electric train in acceleration mode was performed for four variants of the longitudinal profile. Accuracy was evaluated by the absolute deviations of the final values of distance and time from the reference calculation results. The Derringer–Suich method was used to form a composite indicator accounting for accuracy and computation time. Results: the choice of integration method for the computational core of an automatic train operation system should take into account the independent variable and step size. Integration with respect to speed provides high computational performance over intervals of monotonic speed change, but the calculation becomes more complicated when speed is maintained. Integration with respect to time is convenient when the duration of individual driving modes has to be maintained. Since in both cases the path is a dependent variable, changes in slope, speed limits and control commands associated with certain coordinates may occur within a step and must be taken into account separately. When integration is performed with respect to distance, the coordinates of these changes are used directly as step boundaries. For explicit second-order methods used for integration with respect to distance, a broad range of step sizes with a high composite indicator was identified across the profiles considered. Practical significance: the results make it possible to select the mathematical basis for the computational core of an automatic train operation system, taking into account the required balance between accuracy and computational performance, the features of the input dependencies and the applicability range of each method.

KEYWORDS

electric train, automatic train operation, longitudinal motion, numerical integration, derringer-suich method

Text

Text (PDF)

Read Download

1. Vnedrenie sistemy avtovedeniya dlya povysheniya tochnosti upravleniya i energoeffektivnosti vysokoskorostnogo elektropoezda / O. S. Valinskiy [i dr.] // Izvestiya Peterburgskogo universiteta putey soobscheniya. 2025. T. 22, vyp. 1. S. 7–18. DOI:https://doi.org/10.20295/1815- 588X‑2025-1-7-18

2. Yurenko K. I., Fandeev E. I. Principy postroeniya i imitacionnoe modelirovanie sistem avtovedeniya elektropodvizhnogo sostava // Izvestiya YuFU. Tehnicheskie nauki. 2016. № 8 (181). S. 88–102. DOI:https://doi.org/10.18522/2311-3103-2016-8-88-102

3. Yurenko K. I., Savos'kin A. N., Fandeev E. I. Matematicheskoe modelirovanie energooptimal'nyh rezhimov vedeniya poezda s uchetom vozmuscheniy // Izvestiya vysshih uchebnyh zavedeniy. Severo-Kavkazskiy region. Tehnicheskie nauki. 2015. № 3 (184). S. 34–44. DOI:https://doi.org/10.17213/0321-2653-2015-3-34-44

4. Ablyalimov O. S. Uravnenie dvizheniya poezda i nekotorye metody ego resheniya // Universum: tehnicheskie nauki. 2020. № 9 (78). URL: https://7universum.com/ru/tech/ archive/item/10715 (data obrascheniya: 23.07.2026).

5. Ablyalimov O. S. Effektivnost' resheniya differencial'nyh uravneniy vida y′ = f (x, y) metodom hord // Universum: tehnicheskie nauki. 2020. № 9 (78). URL: https://7universum. com/ru/tech/archive/item/10713 (data obrascheniya: 23.07.2026).

6. Butcher J. C. Implicit Runge–Kutta Processes // Mathematics of Computation. 1964. Vol. 18, no. 85. Pp. 50–64. DOI:https://doi.org/10.1090/S0025-5718-1964-0159424-9

7. Kennedy C. A., Carpenter M. H. Diagonally Implicit Runge–Kutta Methods for Ordinary Differential Equations: A Review. NASA/TM-2016-219173. Hampton: NASA Langley Research Center, 2016.

8. Array Programming with NumPy / C. R. Harris [et al.] // Nature. 2020. Vol. 585. Pp. 357–362. DOI:https://doi.org/10.1038/s41586-020-2649-2

9. Derringer G., Suich R. Simultaneous Optimization of Several Response Variables // Journal of Quality Technology. 1980. Vol. 12, no. 4. Pp. 214–219. DOI:https://doi.org/10.1080/00224065.1980.11980968

10. Derringer G. C. A Balancing Act: Optimizing a Product’s Properties // Quality Progress. 1994. Vol. 27, no. 6. Pp. 51–58.

GOST

Valinskiy O. S., Evstaf'ev A. M., Sychugov A. N., Suhanov E. V. Comparative Assessment of Numerical Integration of the Equation of Motion of an Electric Train for Automatic Train Operation // Bulletin of scientific research results. 2026. no. 3. pp. 7-22. DOI: https://doi.org/10.20295/2223-9987-2026-3-7-22. (Date of access 12.10.2026).

MLA

Oleg S. Valinskiy, Evstaf'ev A. M., Sychugov A. N., Suhanov E. V.. "Comparative Assessment of Numerical Integration of the Equation of Motion of an Electric Train for Automatic Train Operation." Bulletin of scientific research results 2026.3 (2026): 7-22. DOI: https://doi.org/10.20295/2223-9987-2026-3-7-22.

APA

Valinskiy O. S., Evstaf'ev A. M., Sychugov A. N., Suhanov E. V. (2026). Comparative Assessment of Numerical Integration of the Equation of Motion of an Electric Train for Automatic Train Operation. Bulletin of scientific research results, 2026(3), 7-22. https://doi.org/10.20295/2223-9987-2026-3-7-22

Chicago

Oleg S. Valinskiy, Evstaf'ev A. M., Sychugov A. N., Suhanov E. V.. "Comparative Assessment of Numerical Integration of the Equation of Motion of an Electric Train for Automatic Train Operation." Bulletin of scientific research results 2026, no. 3 (2026): 7-22. https://doi.org/10.20295/2223-9987-2026-3-7-22.

WIKI

{{cite journal | author = Valinskiy O. S., Evstaf'ev A. M., Sychugov A. N., Suhanov E. V. | title = Comparative Assessment of Numerical Integration of the Equation of Motion of an Electric Train for Automatic Train Operation | journal = Bulletin of scientific research results | year = 2026 | volume = 2026 | issue = 3 | pages = 7-22 | doi = 10.20295/2223-9987-2026-3-7-22 }}

Copy

Export: BibTeX EndNote RefMan RusMARC

COMPARATIVE ASSESSMENT OF NUMERICAL INTEGRATION OF THE EQUATION OF MOTION OF AN ELECTRIC TRAIN FOR AUTOMATIC TRAIN OPERATION Comparative Assessment of Numerical Integration of the Equation of Motion of an Electric Train for Automatic Train Operation

Confirmation