# A Hybrid Model for Freight Train Air Brake Condition Monitoring
## 2 Materials and Methods
### Main Components Implemented in the Model
#### 2.3.4. First Phase Device  
The first phase device (labelled (4) in Figure 6) is a complex component that can be considered as a feature of the relay valve component. It defines a non-linear correction of the brake cylinder pressure depending on the distributor pressure variation. The first braking phase provides a large opening connection between the auxiliary reservoir and the brake cylinder. This device allows the brake cylinder to be rapidly filled with compressed air and to apply nearly 15% of its maximum force [27], keeping the brake in contact with the wheel. The first braking phase ends when the brake cylinder pressure reaches 0.4 bar (identified as Fb in Table 1). While in the literature [27] the first braking phase is modelled through threshold settings tuned from the experimental dataset, causing an increase in the number of model parameters to be identified and a loss of air brake model generalisation, in this paper, the first braking phase is modelled by studying the correlation between the brake cylinder response and the distributor pressure variation caused by the activation of the accelerated chambers, representing thus a novel approach. While most of the models found in the literature are focused on replicating the steady-state condition of the brake cylinder pressure, the simulation of the first braking phase allows to improve the ability of the diagnostic model to recognise malfunctions that are usually very difficult to detect (i.e., push-rod out of adjustment, higher stroke of the brake cylinder piston, variation of the stiffness of the brake cylinder spring and manual brake activation during train moving condition). Since the accelerated chambers are small volume reservoirs, the filling by the main brake pipe determines an instantaneous reduction of the main brake pipe pressure followed by a limited pressure increase derived by a flux-out mechanism, as reported in [34]. Since the distributor is modelled as the main brake pipe depression, we can correlate the activation of the first braking phase with the distributor pressure variation. In Figure 10a,b, a schematic overview of the activation and de-activation of the first braking phase is shown. Figure 9: Transfer function of (**a**) wagon weight distribution and (**b**) the weighing valve. During the first braking phase, the alimentation piston sub-assembly (1) is blocked by the spring compression pre-loading force. If the distributor pressure is not able to overcome the pre-loading condition of the spring, its contribution only affects the upper surface of the alimentation piston (2), determining the opening of the alimentation valve (3) and allowing the air passage from the auxiliary reservoir to the brake cylinder. When the distributor pressure has overcome the spring pre-loading condition, the sub-assembly gets attached to the alimentation piston, determining a non-differential contribution of the distributor pressure since it acts with the same magnitude both at the top and end piston area (as shown by the red arrows in Figure 10b). In this case, the alimentation valve opening is driven by the command pressure and the load-dependent action is started. In Figure 11, the first braking phase peak detection model is shown. The first braking phase model identifies fluctuations in the distributor pressure resulting from the activation of the accelerated chamber, which temporarily reduces the distributor pressure. The model determines \(\mathrm{P_{ref}}\) as the distributor pressure at which the accelerated chamber begins operating, defining \(\mathrm{t_{start}}\) as the activation start time and \(\mathrm{t_{end}}\) as the time when the chamber is fully filled, causing the distributor pressure to exceed the threshold \(\mathrm{p_{ref}}\). In particular, \(\mathrm{p_{ref}}\) is not a fixed parameter in the model; it is automatically detected and varies based on the braking action initiated by the locomotive and the distance between the locomotive and the wagon. Figure 11: First braking phase peak detection.