# A Hybrid Model for Freight Train Air Brake Condition Monitoring
## 3 Results and Discussion
### Discussion  
The relative error between the model and experimental brake cylinder pressure is used as a statistical indicator to assess the accuracy of the results, as shown in Equation (6). \[\text{Relative error}=\frac{1}{n}\sum_{\text{i=1}}^{n}\frac{\left|\text{P}_{ \text{bc model}}(\text{i})-\text{P}_{\text{bc experimental}}(\text{i})\right|}{\left|\text{P}_{ \text{bc experimental}}(\text{i})\right|}.100\% \tag{6}\]  
The error is calculated individually for each phase to evaluate the model's performance in detail. For Datasets 4 and 5, which include multiple braking actions, the overall relative error is calculated as the average of the relative errors for each individual braking action. As illustrated in Figure 19, the model accurately reproduces the experimental brake cylinder pressure, with a maximum relative error of 14% observed during the holding phase of Dataset 4. Figure 19 highlights the statistical variation of the relative error across phases, showing that the mean relative error remains below 10% for all phases. This demonstrates the model's reliability in simulating standard air brake operating conditions. Since the ultimate objective, as outlined in Figure 1, is to use the model for developing diagnostic algorithms, achieving high accuracy in replicating nominal air brake behaviour is a highly encouraging result. In this framework, higher variations in relative errors from the one expected may serve as indicators of potential air brake system malfunctions. Furthermore, the model exhibits good accuracy in reproducing the first braking phase, resulting in a lower mean relative error compared to other phases. To enhance the efficiency of the air brake system diagnostics, the model must accurately simulate not only the brake cylinder pressure but also the timing of each phase. While the model may effectively describe the pressure during the holding phase, it is equally important to accurately estimate the time required to reach that condition. Monitoring the timing of different phases is, in fact, crucial for malfunction identification. For instance, issues with the distributor that regulates the system's filling time could manifest as irregularities in phase timing. To assess this aspect, the relative error in the braking phase time, defined as the time required to reach 90% of the maximum brake cylinder pressure, is calculated. A comparison is made with existing results from the literature [27], which provides a detailed analysis of braking times during emergency braking actions. As shown in Figure 20, the model demonstrates a strong correlation with results reported in [27], highlighting its reliability in capturing both pressure dynamics and phase timing. Figure 19: Relative error comparison between the model and the experimental brake cylinder pressure (**a**) for each Dataset at different phases and (**b**) for all Datasets at different phases. Although the model achieves comparable results to those in [27] when evaluating braking time, it introduces several innovative features and contributions. Notably, the air brake model in this study incorporates critical components such as the weighing valve, relay valve and kink valve, enhancing both its realism and functionality. Additionally, it accounts for the releasing phase by including the brake release valve. A major innovation is the inclusion of the first braking phase device, which captures the dynamics of the accelerated chambers. Another distinctive feature is the evaluation of brake cylinder pressure build-up as a saturation of the brake mode and the reduction of the required model parameters, optimising the computational costs. However, some limitations of the model should still be considered. Since the air brake pipe is not included in the model, the evaluation of the brake cylinder pressure can outline possible malfunctions in other brake system components without providing detailed insights into specific issues.