# State of Health Estimation for Lithium-Ion Batteries Based on TCN-RVM
## 3 Results and Discussion  
### Single-Cell SOH Estimation  
To effectively address the core challenges of capturing long-range dependencies in time-series data and quantifying prediction uncertainties, this study constructs a fused architecture of Temporal Convolutional Network (TCN) and Relevance Vector Machine (RVM). TCN achieves the accurate capture of long-term temporal patterns by virtue of the multi-scale receptive field characteristics of dilated convolution, while RVM provides reliable confidence intervals for prediction results based on the Bayesian probabilistic framework. The two collaborate to form a model with both a long-range dependency modeling capability and uncertainty quantification function.  
In this study, the CS2_35 battery from the University of Maryland dataset was used as the training set, while CS2_36, CS2_37, and CS2_38 batteries were selected as the test sets to train the TCN-RVM model. Figure 6 shows the estimated SOH curves for the three test batteries. As can be observed, the SOH generally exhibits a downward trend with increasing cycle number. The predicted SOH values closely match the reference values for all three batteries, although slight deviations are present.  
To further illustrate the SOH estimation accuracy for the test sets, the error versus cycle number curves are shown in Figure 8. For the CS2_36 and CS2_38 batteries, the error fluctuates within the range of \(-0.04\) to \(0.04\) during the first \(100\) cycles. As the cycle number increases, the fluctuation range narrows to between \(-0.02\) and \(0.02\). In contrast, the CS2_37 battery shows slightly larger error fluctuations, ranging from \(-0.06\) to \(0.06\) within the first \(100\) cycles. The larger fluctuations in the CS2_37 battery may be attributed to its intrinsic properties (such as material or structural differences) or a lower adaptability of the prediction model to this specific battery. Overall, the TCN-RVM model demonstrates high accuracy in SOH estimations, effectively handling data from different battery types. Moreover, as the number of cycles increases, the prediction results tend to stabilize.  
Figure 7: Framework diagram of the prediction process.  
To quantitatively assess the performance of different models, root mean square error (RMSE) and mean absolute error (MAE) were adopted as evaluation metrics for SOH predictions. MAE calculates the average of the absolute differences between the actual and predicted SOH values, reflecting overall prediction accuracy. However, it is less sensitive to extreme values in the dataset. In contrast, RMSE incorporates the square of prediction  
Figure 8: Battery estimation curves and error vs. cycle number curves.  
errors, thereby amplifying the impacts of larger deviations and addressing the sensitivity limitations of MAE. The corresponding formulas are defined as follows:  
\[MAE=\frac{1}{N}{\sum_{i=1}^{N}}{\left|{{y_{i}}-{y_{i}^{*}}}\right|} \tag{15}\]  
\[RMSE=\sqrt{\frac{1}{N}{\sum_{i=1}^{N}}{\left({{y_{i}}-{y_{i}^{*}}}\right)^{2}}} \tag{16}\]  
where \(y_{i}\) denotes the true SOH value of the battery, \(y_{i}^{*}\) represents the estimated value, and \(N\) is the total number of battery cycles in the experiment.  
As shown in Table 4, the MAE and RMSE values for the three test batteries do not exceed 0.75%, indicating the high accuracy and reliability of the proposed model.