  In this paper, we demonstrate the exactness proof for the energy-efficient
train control (EETC) model based on convex optimization. The proof of exactness
shows that the convex optimization model will share the same optimization
results with the initial model on which the convex relaxations are conducted.
We first show how the relaxation on the initial non-convex model is conducted
and provide analysis to show that the relaxations are convex constraints and
the relaxed model is thus a convex model. Subsequently, we prove that the
relaxed convex model will always achieve its optimal solution on the initial
equality constraints and the optimal solution achieved by convex optimization
will be the same as the one obtained by the initial non-convex model and the
relaxations applied are exact. A numerical verification has been conducted
based on a typical urban rail system with a steep gradient. The results of this
paper shed lights on further applications of convex optimization on
energy-efficient train control and relevant areas related to operation and
control of low-carbon transportation systems.
