Energy-optimal Timetable Design for Sustainable Metro Railway Networks

TL;DR

This work addresses energy-efficient timetable design in CBTC-enabled metro networks by formulating a single-stage linear program that simultaneously minimizes total energy consumed during acceleration and maximizes the transfer of regenerative braking energy. Energy modeling is data-driven and affine, enabling real-time prediction of network energy use without time-consuming simulations, while robust box uncertainty safeguards functional constraints. The model demonstrates substantial practical impact on Shanghai Line 8, achieving 20.93%–28.68% energy reductions with sub-second solution times, and is poised for integration into industrial timetable compilers such as the Thales system. Overall, the approach offers a scalable, predictive, and operationally viable framework for sustainable metro scheduling with broad applicability across CBTC-enabled networks.

Abstract

We present our collaboration with Thales Canada Inc, the largest provider of communication-based train control (CBTC) systems worldwide. We study the problem of designing energy-optimal timetables in metro railway networks to minimize the effective energy consumption of the network, which corresponds to simultaneously minimizing total energy consumed by all the trains and maximizing the transfer of regenerative braking energy from suitable braking trains to accelerating trains. We propose a novel data-driven linear programming model that minimizes the total effective energy consumption in a metro railway network, capable of computing the optimal timetable in real-time, even for some of the largest CBTC systems in the world. In contrast with existing works, which are either NP-hard or involve multiple stages requiring extensive simulation, our model is a single linear programming model capable of computing the energy-optimal timetable subject to the constraints present in the railway network. Furthermore, our model can predict the total energy consumption of the network without requiring time-consuming simulations, making it suitable for widespread use in managerial settings. We apply our model to Shanghai Railway Network's Metro Line 8 -- one of the largest and busiest railway services in the world -- and empirically demonstrate that our model computes energy-optimal timetables for thousands of active trains spanning an entire service period of one day in real-time (solution time less than one second on a standard desktop), achieving energy savings between approximately 20.93% and 28.68%. Given the compelling advantages, our model is in the process of being integrated into Thales Canada Inc's industrial timetable compiler.

Figures (8)

• Figure 1: This figure graphically illustrates a train's energy consumption and regeneration cycle in CBTC systems. When a train makes a trip from the origin platform to the destination platform, it transitions through four phases - acceleration, speed holding, coasting, and braking, as shown in the speed vs. time graph at the top. The acceleration phase requires the most amount of energy energy, as denoted by the red-shaded area in the power vs. time graph at the bottom. In contrast, the speed holding and coasting phases entail minimal to zero energy usage. Notably, during the braking phase, the train produces regenerative braking energy, represented by the green-shaded area in the power vs. time graph. With appropriate scheduling, this energy can be strategically transferred to nearby accelerating trains.
• Figure 2: This figure graphically illustrates all the functional constraints present in a metro railway network that the timetable has to satisfy.
• Figure 3: This figure graphically illustrates how the energy consumed by a train $t$ going from platform $i$ to $j$ on track $(i,j)$ varies as the trip time $a_j^t - d_i^t$ is varied. The longer range of trip time, while not practical for a metro railway network, is shown to illustrate the nonlinear nature of the consumed energy versus trip time in long inter-city travels that often span a few hours. On the other hand, the valid range of trip time in a metro network denoted by $[\underline{\tau}_{ij}^{t}, \overline{\tau}_{ij}^{t}]$ is on the order of seconds, and in such a setup an affine approximation for the consumed energy can be reasonable.
• Figure 4: This figure illustrates how to compute the effective braking and acceleration phases of trains using the FWHM (Full Width at Half Maximum) method. Consider a train $t$ arriving (i.e., braking) at, then dwelling, and finally departing (i.e., accelerating) from platform $i$. While the exact power vs. time graph of a train is challenging to model analytically, it is possible to compute a very robust approximation by applying the FWHM method. By doing so, the effective braking and accelerating phase of a train can be compactly represented by these rectangular approximations, which allows us to identify the beginning and end of the effective braking phase of train $t$, denoted by $a_{i}^{t}-\underline{\beta}_{i}^{t}$ and $a_{i}^{t}-\overline{\beta}_{i}^{t}$, respectively, and the beginning and end of the effective accelerating phase of the train $t$, represented by $d_{i}^{t}+\underline{\alpha}_{i}^{t}$ and $d_{i}^{t}+\overline{\alpha}_{i}^{t}$, respectively.
• Figure 5: This figure illustrates all the possible overlapping times $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$. For the first four cases, $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$ shown using the red-shaded region is nonpositive, whereas for the next nine cases $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$ is positive shown using the green-shaded region. The overlapping time $\sigma_{ij}^{t\overset{\rightharpoonup}{t}}$ admits the closed form $\min\left\{ a_{j}^{\overset{\rightharpoonup}{t}}-\overline{\beta}_{j}^{\overset{\rightharpoonup}{t}},d_{i}^{t}+\overline{\alpha}_{i}^{t}\right\} +\min\left\{ -d_{i}^{t}-\underline{\alpha}_{i}^{t},-a_{j}^{\overset{\rightharpoonup}{t}}+\underline{\beta}_{j}^{\overset{\rightharpoonup}{t}}\right\}$.

Theorems & Definitions (3)

• definition 4.1
• definition 4.2
• definition 4.3