# A Hybrid Model for Freight Train Air Brake Condition Monitoring
## 2 Materials and Methods
### Model Flow Chart and Data Pre-Processing  
In Figure 3, a schematic overview of the air brake system model flow chart is shown. In the acquisition stage, the monitoring system (of the same type as the one described in [28]) gathers experimental data from three key points: the main brake pipe, the weighing valve and the brake cylinder. The main brake pipe pressure is then pre-processed to identify the time intervals for braking, holding and releasing phases. These phase durations, along with the main brake pipe pressure, weighing valve pressure and the model parameters, serve as inputs to the air brake model, which outputs the expected brake cylinder pressure. The model validation is then performed by comparing the experimental and the model brake cylinder pressure. Any deviation between the modelled and experimental brake cylinder pressures can indicate potential malfunctions within the system, as illustrated in Figure 1.  
Figure 2: T3000e wagon air brake plant scheme. The brake cylinder response is influenced by several factors: the main brake pipe pressure, which enables the simulation of both service and emergency braking; the weighing valve pressure, which allows for the simulation of different loading conditions; the model parameters, which allow testing different brake system configurations and brake modes (freight or passenger). Before the main brake pipe pressure is introduced as input in the model, it undergoes a two-step pre-processing phase: peak detection and phase classification. During peak detection, the time history of the main brake pipe pressure, recorded during acquisition, is divided into distinct braking actions, as shown in Figure 4. The time history of the main brake pipe pressure is first filtered using a moving average filter to reduce noise. Then, the time derivative of the filtered signal is calculated. The minima peaks in the derivative correspond to the start of the brake action, indicating moments of instantaneous pressure reduction in the main brake pipe, as shown in Figure 4. The initial pressure value of the main brake pipe at the start of each braking action is then determined, which is crucial because this value is held constant in the control reservoir and regulates the pressure transmitted to the distributor. A phase classification algorithm is then used to identify three main braking conditions: braking, releasing and holding. The braking phase begins with a decrease in the main brake pipe pressure, triggering the distributor to feed air into the brake cylinder, increasing its pressure until stabilisation. The holding phase follows, where both the main brake pipe and brake cylinder pressures remain constant, maintaining the braking force. Finally, the releasing phase starts when the main brake pipe pressure rises, signalling the distributor to vent the brake cylinder, reducing its pressure and fully releasing the brakes. The algorithm  
Figure 4: Time histories of (**a**) the main brake pipe peak detection algorithm and (**b**) the main brake pipe partition. Figure 3: Air brake system model flow chart overview. is based on the distributor pressure, calculated as the depression between the main brake pipe pressure (Figure 5a) and the control reservoir pressure: this signal is passed through a moving average filter to cut off the fast dynamics related to the accelerated chambers activation (Figure 5b, \(\mathrm{P}_{\mathrm{filter}}\)) and then a first-order differential function is applied (Figure 5c, \(\mathrm{P}_{\mathrm{filter}}\)). The braking condition starts when \(\mathrm{P}_{\mathrm{filter}}\) is higher than 0.005 bar/s, while the releasing is the condition where \(\mathrm{P}_{\mathrm{filter}}\) is lower than \(-0.005\) bar/s, and the holding is the intermediate condition, as shown in Figure 5d. The information regarding the phase partition and classification is introduced inside the model. A hybrid air brake model using empirical and fluid-dynamic simplified equations is used to evaluate the brake cylinder pressure.