Optimal driving strategies for a fleet of trains

TL;DR

The study addresses reducing fleet energy costs under peak-demand constraints while preserving journey times. It adopts constrained optimal control, leveraging the Pontryagin framework to derive a structured, switching-optimal policy where each train uses an unconstrained speed $V_j$ on unconstrained intervals and constrained speeds $V_{j,k}$ on active energy-constraint intervals, linked by $1 + w_k = \varphi'(V_j)/\varphi'(V_{j,k})$. The authors formulate a fleet-wide objective with interval energy caps ${\mathcal Q}_k$, derive the Hamiltonian, Lagrangian, adjoint equations, KKT conditions, and explicit control laws, and show how adjoint evolution governs interval transitions. They demonstrate feasibility and practical applicability on fleets (3 trains with one constraint, 5 with three consecutive constraints) and discuss integration into onboard DAS like Energymiser, with prospects for extension to tracks with gradients and broader incentive-based deployment.

Abstract

In order to manage electricity transmission and distribution it is now common practice for system operators to offer financial incentives that encourage large consumers to reduce energy usage during designated peak demand periods. For train operators on large rail networks it may be profitable -- with selected individual journeys -- to reduce energy usage during peak times and increase energy usage at other times rather than simply minimizing overall energy consumption. We will use classical methods of constrained optimization to find optimal driving strategies for a fleet of trains subject to limits on total energy consumption during specified intermediate time intervals but with no change to individual journey times. The proposed strategies can be used by a large rail organisation to reduce overall operating costs with only minimal disruption to existing schedules and with no changes to important departure and arrival times.

Figures (5)

• Figure 1: Optimal speed profiles $v_1 = v_1(t)$ in Example \ref{['ex:1']}.
• Figure 2: Optimal speed profiles for train ${\mathfrak T}_1$ when $t \in [t_1,t_2] = [750, 1350]$ in Example \ref{['ex:1']} with ${\mathcal{Q}}_1 = 100 M, 200 M, 300 M$. When the restricted optimal driving speed $V_{1,1}$ increases the speed $W_{1,1} = v_1(t_1)$ decreases ($\partial W_{1,1}/\partial V_{1,1} < 0$) and the speed $W_{1,2} = v_1(t_2)$ increases ($\partial W_{1,2}/\partial V_{1,1} > 0$). The duration $\Delta \tau_{\,1,1}$ of the speedhold phase also increases. In this case $E_1 = M \Delta \tau_{\,1,1} \varphi(V_{1,1}) = {\mathcal{Q}}_1$.
• Figure 3: Graphs of the optimal speeds $v_1$ against the modified adjoint variable $\eta_1$ for train ${\mathfrak T}_1$ when $t \in [t_1,t_2] = [750, 1350]$ in Example \ref{['ex:1']} with ${\mathcal{Q}}_1 = 100 M, 200 M, 300 M$ showing (i) the optimal unrestricted driving speeds $V_1$, (ii) the optimal restricted driving speeds $V_{1,1}$, (iii) the switching speeds $W_{1,1} = v_1(t_1)$ and $W_{1,2} = v_1(t_2)$, (iv) the $(\eta_{1,a},v_1)$ curves for the maximum acceleration phases on $(0,t_1)$ and $(t_2,T)$ and the associated minimum turning points for $\eta_{1,a}$ at $(1, V_1)$ and (v) the $(\eta_{1,c},v_1)$ curves for the coast phases on $(t_1,t_2)$ and the associated maximum turning points for $\eta_{1,c}$ at $(1+w_1, V_{1,1})$.
• Figure 4: Optimal speed profiles for trains $\{ {\mathfrak T}_j\}_{j=1}^3$ in Example \ref{['ex:2']}.
• Figure 5: Optimal speed profiles for trains $\{ {\mathfrak T}_j\}_{j=1}^5$ in Example \ref{['ex:3']}.

Theorems & Definitions (4)

• Remark 1
• Example 1
• Example 2
• Example 3