Energy replenishment optimization for inland liner electric container ships using Benders decomposition

Achieving net-zero greenhouse gas (GHG) emissions has become a central priority for the maritime transportation sector, with stricter emission reduction regulations accelerating the adoption of zero-emission technologies, including full electrification of inland waterways. These waterways, with their specific operational profiles and voyage distances, make battery-powered vessels a feasible option for decarbonization. However, electric vessels face several operational challenges, including limited battery capacity, unevenly distributed charging and swapping facilities, and grid capacity constraints. Together with resource competition from vessels arriving simultaneously, these factors create a bottleneck for continuous energy replenishment along fixed liner services, hindering the large-scale adoption of electric ships. To address this challenge, this study focuses on a fixed inland container liner service in the Yangtze River Delta. It develops an optimization framework for multi-vessel, multi-port, and multi-technology energy replenishment from the perspective of shipping operators. The framework incorporates both shore-side fast charging and containerized battery swapping, and dynamically models state-of-charge (SOC) evolution across a sequence of ports under multiple resource constraints. Based on this framework, a mixed-integer programming model is formulated to determine the optimal replenishment schedule for the electric liner fleet. A Benders decomposition algorithm is further designed to solve the model, enabling an iterative master-subproblem structure. Case studies and numerical experiments of different scales verify the model’s effectiveness and the solution approach. Scenario analyses further evaluate how replenishment modes, port facility configurations, vessel endurance levels, the number of serviceable ports, and maximum charging power impact replenishment costs, service feasibility, and operational performance. The findings provide insights to support the reliable and cost-effective deployment of electric container vessels, which is an essential component in advancing inland shipping toward net-zero GHG emissions.