Joint Optimization of Energy Replenishment and Sailing Speed for Inland Electric Vessels under Time-of-Use Pricing

Battery-powered propulsion offers a pathway for reducing inland shipping emissions. However, low battery energy density limits sailing range and may require energy replenishment during a voyage, thereby complicating energy and voyage planning for inland electric vessels. Time-of-use (TOU) pricing creates cost-saving opportunities but further complicates planning because energy replenishment and sailing speed are closely coupled. This study develops a joint optimization framework for an inland electric vessel under TOU pricing that determines replenishment ports, technologies, amounts, and leg-specific sailing speeds to minimize total replenishment costs. The problem is formulated as a mixed-integer nonlinear programming model and reformulated as a mixed-integer linear programming approximation through equivalent linearization and speed discretization. A Yangtze River case study with five operating conditions evaluates the proposed framework. The results show that, under the proposed framework, TOU pricing reduces total replenishment costs by 42.4–43.4% compared to fixed pricing. Relative to a sailing-speed optimization benchmark, joint optimization under TOU pricing reduces replenishment costs by 9.8–12.5% and energy consumption by 4.2–5.6%. The cost-saving potential also varies with the voyage time limit, voyage start time, relative charging and battery swapping rates, and the availability of opportunity charging.