Eco-Efficient Vessel Dynamics: An Interval Approach to Heave Response and Energy-Saving in Rough Seas

Maritime transport faces increasing pressure to reduce fuel consumption and emissions, yet vessel performance under variable sea states remains difficult to bound reliably. Traditional stochastic and data-driven models provide probabilistic forecasts but lack strict guarantees in extreme or out-of-sample conditions. This study develops a deterministic arithmetic-interval framework that replaces uncertain hydrodynamic parameters and wave forcing with bounded intervals. The vessel’s single-degree-of-freedom heave equation is reformulated as an interval differential equation, and existence and uniqueness of the resulting solution tube are established. Validated numerical techniques-interval Taylor expansions, Picard iteration, and adaptive subdivision-are used to compute tight heave envelopes. An interval energy metric integrates worst-case power demand over a voyage, and a branch-and-bound global optimizer selects control parameters (e.g., speed schedules) that minimize the upper-bound energy while satisfying seakeeping constraints. Two hypothetical Karnataka-coast scenarios (“calm” and “rough” seas) demonstrate the rigor and efficiency of the approach. Computed energy-consumption intervals exactly enclose corresponding Monte-Carlo extremes, confirming tightness without large sample sizes. Rough-sea conditions increase worst-case energy demand by approximately 75% despite negligible heave amplitudes at the micron scale. Sensitivity analysis shows that wave-amplitude uncertainty dominates energy variability, while vessel stiffness and damping have minimal influence. The proposed interval framework eliminates under-coverage of worst-case energy (0% missed extremes) and remains within 3–6% of the tightest Monte-Carlo 99% confidence bands, achieving comparable bound tightness with two orders of magnitude fewer model evaluations than CNN–BiLSTM–Attention and kernel-density-based predictors. Benchmarking against linear heave RAO predictions confirms hydrodynamic consistency. The approach provides decision-makers with mathematically guaranteed bounds, supporting targeted measurement, control, and sustainable maritime operations.