Deep learning-based virtual sensing of electrical responses in free-piston linear generators via piston dynamics

Free-piston linear generators are compact multi-fuel power units, but the strong nonlinear coupling between piston motion and electrical output makes the kinematic-electrical mapping difficult to model and limits power-demand-oriented motion design. This study develops a deep-learning-based virtual sensing framework that predicts three-phase voltage, three-phase current, and output power from piston displacement, velocity, acceleration, and load resistance. The framework is trained and tested on experimental data from a test prototype under nine load conditions using a leave-one-condition-out strategy for each load. Four deep-learning models are evaluated, including two standalone models and two combined architectures. Across all load conditions, the bidirectional long short-term memory network achieves mean coefficients of determination above 0.999 for voltage, current, and power, together with the lowest prediction errors, which indicates accurate reconstruction of global waveforms. The average relative errors of peak voltage, peak current, and peak power are 2.43 %, 2.79 %, and 1.36 %, so all peak-value errors remain below 3 %. This network is further compared with two baseline methods reported in previous studies under the same data partitions, and it consistently achieves higher coefficients of determination and lower prediction errors in all cases. These results demonstrate that the proposed framework is an effective tool for virtual sensing of electrical parameters and provides a practical basis for future power-oriented inverse design of piston motion profiles.