# PEMFC RUL Prediction for Non-Stationary Time Series Based on Crossformer Model
## 2 Problem Formulation  
This section will define concepts related to PEMFC lifetime prediction modeling and formalize the relevant research questions. Definition 1: The number of occurrences of fixed-range current values during PEMFC operation is statistically calculated. The range with the highest occurrence is defined as the steady-state current value, as shown in Equation (1). \[I_{\text{steady}}=MAX(f_{\text{1}}) \tag{1}\]  
where \(I_{\text{steady}}\) represents the steady-state current and \(f_{\text{1}}\) denotes the frequency of occurrence of the current during operation. Figure 1 shows the frequency of different currents during the operation of the logistics vehicle's PEMFC, derived through mathematical statistical methods. The results indicate that currents in the range of 10-100 A occur less frequently, likely because the vehicle relies mainly on the battery for discharge during these conditions, and the fuel cell is either not operating or generating only minimal output. Additionally, while the current range of 100-110 A exhibits a relatively high occurrence frequency, it is more likely associated with low-speed or light-load operating conditions, where the fuel cell output is relatively unstable and less representative of overall system operation. Therefore, despite its high frequency, this range is not suitable for use as the steady-state current. In contrast, the current range of 198.4-199.4 A exhibits the highest occurrence frequency and better represents the stable output of the fuel cell under primary operating conditions. Thus, it is identified as the steady-state current of the PEMFC installed in the logistics vehicle. Figure 1: Frequency of currents in the PEMFC installed in the logistics vehicle during operation. **Definition 2**. : _The voltage corresponding to the steady-state current during operation is defined as the steady-state voltage, as shown in Equation (2)._  
\[\text{U}_{steady}=\text{U}\Big{(}\text{I}=\text{I}_{steady}\Big{)} \tag{2}\]  
_where \(\text{U}_{steady}\) represents the steady-state voltage. Figure 2 illustrates the steady-state voltage values of the PEMFC installed in the logistics vehicle, with 198.4-199.4 A as the steady-state current, along with linear fitting. The primary purpose of this linear fitting is to verify the stability of the steady-state voltage. If the fitting results show significant fluctuations or an unclear trend, they may indicate instability or abnormalities in the operational state of the fuel cell. In such cases, it would be necessary to first address any anomalies or unstable data. However, as observed from the results in Figure 2, the steady-state voltage shows relatively stable variations, thus making it suitable for subsequent predictions of remaining useful life._  
**Definition 3**. : _In this study, the steady-state voltage of the PEMFC during operation is adopted as the evaluation criterion for the degradation of remaining useful life, based on research findings of the demonstrated operation of various fuel cell vehicles in typical regions conducted by Tsinghua University. It is utilized as the label value for the dataset used in this study._  
When a PEMFC is initially activated for use, it undergoes an activation stage. After this stage, the steady-state voltage of the PEMFC will reach a peak, known as the initial voltage. This initial voltage represents 100% of the battery's life. Following the definition of failure in lithium-ion batteries [17], this paper considers a decrease in steady-state voltage to 80% of the initial voltage as a sign of PEMFC performance failure, indicating the depletion of its useful life and the need for timely shutdown and replacement. **Problem 1**. : _The essence of PEMFC RUL prediction is to predict the steady-state voltage values \(y_{T+1:T+\tau}\in\mathbb{R}^{\tau\times 1}\) within a specified future time window, given a historical time series \(x_{1:T}\in\mathbb{R}^{T\times D}\)[18].