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H.G. Bock¹, R.W. Longman², S. Sager³ and J.P. Schlöder¹

¹Interdisciplinary Center for Scientific Computing (IWR), Ruprecht-Karls-Universität Heidelberg, Germany
²Department of Mechanical Engineering, Columbia University, New York, USA
³Faculty of Mathematics, Otto-von-Guericke-Universität Magdeburg, Germany

Keywords: cam controlled subway trains, energy optimal subway operation, mixed integer optimal control problems, Pontryagin's Maximum Principle, Direct Multiple Shooting, nonlinear optimization boundary value problems, numerical methods.

Summary

The authors have previously developed minimal energy control strategies for the New York Metropolitan Transit Authority. Energy optimal operation of cam controlled subway trains requires the numerical solution of state and control constrained optimal control problems with both continuous and integer controls. The paper describes two rigorous mathematical solution approaches and gives numerical results for a representative station to station ride. The indirect approach based on Pontryagin's Maximum Principle and the Competing Hamiltonians algorithm leads to intricate multi-point boundary value problems in state and adjoint variables with jumps and switching conditions. A direct approach based on outer convexification, relaxation and the Krein-Milman theorem allows for offline solution of mixed integer control problems with no integer gap while avoiding the combinatorial explosion of computing time. Moreover, arbitrarily good approximations by integer solutions with finitely many switches can be constructed by adequate rounding procedures. An advanced Multiple Shooting method for the numerical solution of the resulting (optimization) boundary value problems is described.

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Front Page Browse CCP CSETS CTR IJRT Other Authors Search Purchase Guide FAQ Contact us	Civil-Comp Proceedings ISSN 1759-3433 CCP: 104 PROCEEDINGS OF THE SECOND INTERNATIONAL CONFERENCE ON RAILWAY TECHNOLOGY: RESEARCH, DEVELOPMENT AND MAINTENANCE Edited by: J. Pombo Paper 243 Optimal Control Methods for Energy Optimal Subway Operation H.G. Bock¹, R.W. Longman², S. Sager³ and J.P. Schlöder¹ ¹Interdisciplinary Center for Scientific Computing (IWR), Ruprecht-Karls-Universität Heidelberg, Germany ²Department of Mechanical Engineering, Columbia University, New York, USA ³Faculty of Mathematics, Otto-von-Guericke-Universität Magdeburg, Germany doi:10.4203/ccp.104.243 purchase the full-text of this paper Full Bibliographic Reference for this paper , "Optimal Control Methods for Energy Optimal Subway Operation", in J. Pombo, (Editor), "Proceedings of the Second International Conference on Railway Technology: Research, Development and Maintenance", Civil-Comp Press, Stirlingshire, UK, Paper 243, 2014. doi:10.4203/ccp.104.243 Keywords: cam controlled subway trains, energy optimal subway operation, mixed integer optimal control problems, Pontryagin's Maximum Principle, Direct Multiple Shooting, nonlinear optimization boundary value problems, numerical methods. Summary The authors have previously developed minimal energy control strategies for the New York Metropolitan Transit Authority. Energy optimal operation of cam controlled subway trains requires the numerical solution of state and control constrained optimal control problems with both continuous and integer controls. The paper describes two rigorous mathematical solution approaches and gives numerical results for a representative station to station ride. The indirect approach based on Pontryagin's Maximum Principle and the Competing Hamiltonians algorithm leads to intricate multi-point boundary value problems in state and adjoint variables with jumps and switching conditions. A direct approach based on outer convexification, relaxation and the Krein-Milman theorem allows for offline solution of mixed integer control problems with no integer gap while avoiding the combinatorial explosion of computing time. Moreover, arbitrarily good approximations by integer solutions with finitely many switches can be constructed by adequate rounding procedures. An advanced Multiple Shooting method for the numerical solution of the resulting (optimization) boundary value problems is described. purchase the full-text of this paper (price £20) go to the previous paper go to the next paper return to the table of contents return to the book description purchase this book (price £65 +P&P)
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