# PEMFC RUL Prediction for Non-Stationary Time Series Based on Crossformer Model
###### Abstract  
Proton-Exchange Membrane Fuel Cells (PEMFCs), as efficient and environmentally friendly energy conversion devices, have wide application potential in areas such as transportation, mobile power, and distributed energy. However, the remaining useful life (RUL) issue of PEMFCs has been one of the main challenges limiting their commercialization. The RUL prediction problem of PEMFCs exhibits characteristics of time series forecasting, but its data possess multidimensional features and non-stationarity, which limits the applicability of classical time series forecasting models like the Transformer in solving the RUL prediction problem. In this paper, we propose a PEMFC RUL prediction model based on the Crossformer for non-stationary time series (De-stationary-Crossformer). Firstly, the overall architecture adopts the Crossformer model to extract dependencies between different features and temporal dependencies. Secondly, adaptive normalization is applied to the data to mitigate the non-stationarity in the original data, thereby increasing their predictability. Subsequently, a non-stationary attention mechanism is introduced in the model to simultaneously utilize the non-stationarity in the original data when extracting deep information. Additionally, manual features are introduced through mathematical statistics to enhance the predictive performance of the model. During the training process, the TILDE-Q loss function is used to focus on the similarity between the predicted sequence and the true sequence. The model proposed in this paper improves the MSE by 31% compared to the Transformer and 23% compared to the Crossformer in the experimental prediction of the RUL of PEMFCs in actual vehicles. P remark  
equipped with fuel cells, selecting the appropriate voltage decay to represent the RUL of fuel cells is also an urgent problem that needs to be addressed. In recent years, many researchers have devoted themselves to the prediction of the steady-state voltage of PEMFCs. Research methods can be divided into two categories: model-driven and data-driven. [PERSON] et al. [3] used a two-stage hybrid prediction method to predict the RUL of PEMFCs. In the first stage, the long-term degradation trend in PEMFCs was predicted using evolutionary algorithms and neuro-fuzzy inference methods. The neuro-fuzzy inference method employed was adaptive. In the second stage, based on the data from the first stage, a semi-empirical degradation model of PEMFCs and an adaptive Kalman filter algorithm were combined for RUL prediction. The feasibility of this method was demonstrated through experiments. [PERSON] et al. [4], while retaining the simplicity of the original gray model and the rapid prototyping design, proposed a new type of error-corrected gray prediction model using error correction factors to eliminate the inherent errors of the original gray model. This method demonstrated excellent effectiveness and robustness in aging experiments of lithium batteries and fuel cells and could achieve relatively accurate predictions even over longer time steps. [PERSON] et al. [5] proposed predicting the RUL of PEMFCs using Convolutional Neural Network-Long Short-Term Memory (CNN-LSTM) and Convolutional Neural Network-Bidirectional Long Short-Term Memory (CNN-BiLSTM). They found that the addition of convolutional networks effectively improved the prediction accuracy of LSTM networks. [PERSON] et al. [6], noting the suboptimal long-term predictive performance of LSTM neural networks, proposed Navigation Sequence-Driven LSTM (NSD-LSTM). This method demonstrated high accuracy in long-term prediction capabilities on aging experimental data from two different types of PEMFCs and under different operating conditions. The prediction of the steady-state voltage of PEMFCs falls under the realm of time series forecasting. Recently, in the field of time series forecasting, [PERSON] et al.