[Paper Review] A fast-solved model for energy-efficient train control based on convex optimization

Minling Feng, Kunpeng Wu+1 more|arXiv (Cornell University)|Jan 26, 2022

Railway Systems and Energy Efficiency4 citations

TL;DR

This paper proposes a convex optimization-based model for energy-efficient train control (EETC) that enables fast, globally optimal solutions by relaxing nonconvex constraints related to speed and kinetic energy. The method achieves superior computational efficiency compared to MILP and Radau pseudospectral methods, with numerical validation confirming solution exactness and negligible deviation in relaxed variables.

ABSTRACT

In modern rail transportation, energy-efficient train control (EETC) is concerned with the optimal train speed trajectory or control strategies to achieve the minimum energy cost under various operation and traction constraints. This paper proposes an EETC model based on convex optimization so that the model can be rapidly solved by convex optimization algorithms. The high computational efficiency and robustness of the convex model can be verified by comparing the results achieved by the method proposed by this paper and other mainstream mathematical programming methods including mixed-integer linear programming (MILP) and Radau pseudospectral method (RPM). Based on the characteristics of convex optimization, the proposed method boasts more significant advantages over its counterparts in terms of computational efficiency in the promising online applications for automatic train control systems of various types of rail transportation.

Motivation & Objective

• To develop a computationally efficient and globally optimal solution for energy-efficient train control (EETC) under real-world operational constraints.
• To address the computational bottleneck of existing EETC methods by reformulating nonconvex constraints into a convex framework.
• To verify the exactness of the relaxed convex model by ensuring its optimal solution lies within the feasible region of the original nonconvex problem.
• To demonstrate the superiority of convex optimization in computational speed, especially as problem scale increases, compared to MILP and Radau pseudospectral methods.
• To enable practical online application of EETC in automatic train control systems through fast-solvable, robust optimization.

Proposed method

• The EETC problem is reformulated as a convex optimization model by applying relaxation and convexification techniques to nonconvex constraints involving reciprocal speed (1/v_i) and squared speed (v_i^2).
• Auxiliary variables α_i and β_i are introduced to replace 1/v_i and v_i^2, respectively, transforming the nonconvex terms into convex ones through relaxation.
• The model leverages convex optimization algorithms, enabling rapid solution via interior-point methods or barrier methods, ensuring high computational efficiency and robustness.
• The relaxation is validated by enforcing equality between auxiliary and original variables at optimality, confirming the solution's feasibility in the original problem.
• Numerical comparisons are conducted with MILP and Radau pseudospectral method (RPM), using identical problem instances and verification simulations to ensure fair evaluation.
• A dedicated verification simulation program is implemented to recompute energy consumption and running time using the same speed trajectories, accounting for drag, kinetic energy, and potential energy changes with efficiency factors.

Experimental results

Research questions

• RQ1 Can nonconvex EETC constraints involving 1/v_i and v_i^2 be effectively relaxed and convexified without sacrificing solution accuracy?
• RQ2 Does the resulting convex optimization model maintain exactness, such that its optimal solution lies within the feasible region of the original nonconvex problem?
• RQ3 How does the computational performance of the convex model compare to MILP and Radau pseudospectral method (RPM) as the problem size increases?
• RQ4 Can the convex model achieve faster solution times than MILP and RPM while maintaining comparable solution quality in energy efficiency and trajectory accuracy?
• RQ5 What is the impact of relaxation on the optimality gap, and can the deviation between original and auxiliary variables be proven negligible in practice?

Key findings

• The optimal solution of the convex model lies within the feasible region of the original nonconvex problem, confirming the exactness of the relaxation approach.
• Maximum deviations between original variables (1/v_i, v_i^2) and their auxiliary variables (α_i, β_i) were 0.00074% and 0.00132%, respectively, indicating negligible error.
• Convex optimization achieved significantly faster computation times than both MILP and RPM, especially for large-scale problems (N > 4550), with a slower growth rate in CPU time as problem size increased.
• For CPU time under 100 seconds, the MILP method showed a faster increase in computation time with problem size compared to convex optimization, highlighting the latter’s scalability.
• The energy consumption and running time computed via the verification simulation program confirmed that the convex model’s solution is consistent with physical and operational constraints.
• The convex model outperforms MILP and RPM in computational efficiency while maintaining solution quality, making it highly suitable for online automatic train control applications.