Fault diagnosis and uncertainty quantification for the shaft system of hydropower units based on center-frequency positional encoding and variational Bayesian transformer

Abstract Fault samples of hydropower unit shaft systems can hardly cover all conditions, so unknown states outside the training distribution inevitably appear in online monitoring data. Most existing deep diagnostic models are deterministic: they output a single class label, and can neither characterize the credibility of their own conclusions nor exploit the physical prior of the frequency bands occupied by the modal components. This paper therefore proposes MSEF-BayesFormer, a probabilistic diagnosis model integrating multi-scale entropy features into a variational Bayesian Transformer. A multi-scale entropy feature convolutional embedding module is proposed, in which the refined composite multiscale permutation entropy matrix of the modal components obtained by variational mode decomposition, instead of raw-waveform patches, serves as the input and is mapped into a token sequence by depthwise separable convolution, suppressing at the source the interference of noise on the representation and making fault patterns well separable before they enter the attention interaction. A center-frequency positional encoding is proposed, in which the normalized center frequency of each intrinsic mode function modulates the angular frequency of the sinusoidal positional encoding of the Transformer, explicitly injecting frequency-band physical mechanism information into the model. The attention weights are extended from point estimates to variational Gaussian distributions and trained end-to-end via the evidence lower bound; Monte Carlo sampling yields the predictive distribution, and the predictive uncertainty is decomposed into aleatoric and epistemic components. On a rotor test rig with four condition classes, the average five-fold cross-validation accuracy reaches 0.9523, 2.48 percentage points higher than the deterministic Transformer; the area under the risk–coverage curve is 0.40, better than 0.64 of Deep Ensemble; and the average AUROC of leave-one-class-out unknown-state detection is 0.8139, exceeding MC Dropout and Deep Ensemble by 0.13 and 0.18. In the hydraulic unbalance case at the SK hydropower station, the dominant frequency band on which the model relies is locked onto the blade passing frequency, consistent with the excitation mechanism of this fault.