Convex Formulation of the Zero Emission Vessel Route Planning Problem

TL;DR

The paper tackles the zero emission vessel route planning problem by recasting a nonlinear, nonconvex optimization into a log-convex framework, enabling fast and exact solutions without discretizing vessel designs or sailing speeds. By transforming variables via log-space and leveraging posynomial/log-sum-exp structures, the authors solve a sequence of convex subproblems and recover the original optimal design and operation variables exactly. The approach supports continuous sailing speeds, realistic vessel-design submodels, and tight integration of network and hull dynamics, with an open-source implementation released for reproducibility. Practically, the method enables rapid design sweeps, robust fleet planning, and effective consideration of demand scenarios and service levels for battery-electric shipping across fixed routes.

Abstract

This paper focuses on the zero emission vessel route planning problem, which deals with cost-effective planning of battery-electric vessel services for predetermined routes. Vessel characteristics (including battery capacity), fleet size, cyclic schedule frequencies, sailing leg speeds, and shore charging infrastructure are jointly optimized. The problem is nonlinear and nonconvex in its original form, which makes it intractable for most real-world instances. The conventional approach in the literature is to solve a linear approximation by restricting vessel designs and sailing leg speeds to a small finite set. Contrary to the conventional linearization approach, this paper deals with the nonlinearities directly. We show that the problem exhibits a hidden convex structure uncovered by nonlinear changes of variables. By exploiting the favorable convex form of the transformed problem, we solve it in a few seconds using a free off-the-shelf solver that requires no initial guesses, variable bounds, or parameter tuning. We then easily recover the exact solution to the original nonconvex problem by reversing the variable changes. We provide an open-source implementation of our method.

Figures (14)

• Figure 1: Feasible set of a log-convex optimization problem with two variables. The left panel shows the original nonconvex problem and the right panel shows the log-transformed convex problem. The solid red cross is the optimal solution.
• Figure 2: A route is a sequence of sailing legs between linearly ordered ports from 1 to $N$ and indexed by $i$ and $j$. The figure illustrates a shipping network consisting of three routes constructed from subsets of the same set of $N$ ports. If two or more routes include a pair of ports, multiple services can supply the cargo demand between these ports.
• Figure 3: Line drawing of a hull defined by offset functions. The lines represent the intersection of the hull with transverse planes. Lines on the right-hand side of the vertical centerline represent the forward section, and the left-hand side the aft section.
• Figure 4: Superstructure length as a design variable. Longer superstructure increases passenger capacity, but also electricity consumption, steel weight, construction cost, and port charges.
• Figure 5: Shifting of center of buoyancy due to heel. The lever arm GZ between buoyancy and weight vectors must be positive by design. Under this condition, the vessel reverts to upright orientation.