Energy-efficient train control: from local convexity to global optimization and uniqueness A. R. Albrecht a , P. G. Howlett a,⋆ , P. J. Pudney a , Xuan Vu a,⋆⋆ . a Scheduling and Control Group, Centre for Industrial and Applied Mathematics, Mawson Lakes Campus, University of South Australia, Mawson Lakes, Australia, 5095 Abstract The optimal driving strategy for a train is essentially a power-speedhold-coast-brake strategy unless the track contains steep grades in which case the speedhold mode must be interrupted by phases of power for steep uphill sections and coast for steep downhill sections. The Energymiser R  device is used on freight and passenger trains in Australia and the United Kingdom to provide on-board advice for drivers about energy-efficient driving strategies. Energymiser R  uses a specialized numerical algorithm to find optimal switching points for each steep section of track. Although the algorithm finds a feasible strategy that satisfies the necessary optimality conditions there has been no direct proof that the corresponding switching points are uniquely defined. We use a comprehensive perturbation analysis to show that a key local energy functional is convex with a unique minimum and in so doing prove that the optimal switching points are uniquely defined for each steep section of track. Hence we also deduce that the global optimal strategy is unique. We present two examples using realistic parameter values. Key words: optimal train control; energy minimization; calculus of variations. 1 Background 1.1 Review of optimal train control A comprehensive review of the modern theory of optimal train control can be found in [2,3,5,7–10] and references contained therein. The problem is to minimize the en- ergy required to drive a train from one station to the next within a given time. The optimal strategy is essentially a power–speedhold–coast–brake strategy except that the singular speedhold control phase must be interrupted by phases of regular control to negotiate steep grades. Thus we must insert phases of power for steep uphill sections and coast for steep downhill sections. Hence the optimal strategy becomes an optimal switching strategy. By con- sidering the necessary conditions for optimal switching Howlett et al. [8] showed that the switching points can be determined for each steep section by minimising an ⋆ Corresponding author P. G. Howlett. ⋆⋆ A preliminary paper [1] with a proposed perturbation anal- ysis but no proof of uniqueness was presented by Xuan Vu at the American Control Conference ACC 2011, San Francisco. Email addresses: amie.albrecht@unisa.edu.au (A. R. Albrecht), phil.howlett@unisa.edu.au (P. G. Howlett), peter.pudney@unisa.edu.au (P. J. Pudney), xuan.vu@unisa.edu.au (Xuan Vu). associated local energy functional. Specialized numeri- cal control algorithms developed by the Scheduling and Control Group (SCG) at the University of South Aus- tralia for TTG Transportation Technology are an es- sential component of the Energymiser R  system which provides on-board advice to train drivers about energy- efficient driving strategies. The system is used by ma- jor rail operators in Australia and the United Kingdom. The original algorithms were written in the lazy func- tional programming language Haskell 1 and will com- pute an optimal speed profile for a journey of several hundred kilometres in a few seconds on a standard laptop computer. The algorithms use problem-specific equa- tions and analytic formulæ for key control functions to determine optimal switching points. More information about Energymiser R  can be obtained from the TTG Transportation Technology website 2 . General methods of computational control could conceivably be used for this task but specialized algorithms are more accurate and more efficient [8]. We refer to [8] for a brief review of general numerical control methods. Such methods are not suitable for on-board calculation of optimal driving strategies in real-time. 1 See www.haskell.org/haskellwiki/Haskell 2 See www.ttgtransportationtechnology.com Preprint submitted to Automatica 5 February 2013