Development of an Operational Digital Twin of a Freight Car Braking System for Fault Diagnosis

In the evolving landscape of digital technology, the concept of digital twinning has emerged as one of the most important innovations. Digital twinning, characterized by a one‐to‐one correlation with physical structures in a digital environment, has become an important tool in transferring physical dynamics into digital replicas. This platform, integrating advanced sensor technologies and analytical tools, enables a comprehensive monitoring, diagnosis, and optimization concept. The present research focuses on the application of digital twinning in addressing the complex challenge of managing fault types in freight car brakes. A Digital Twin (DT) model of a freight car braking system is developed that forms the basis for an innovative approach to fault classification. This approach leverages the ability of the DT to emulate and analyze the behavior of the braking system under different conditions. The effectiveness of the proposed method is evaluated through extensive testing and analysis with an experimental data set. The results show that the method is able to accurately diagnose different types of faults. This represents a significant advance in the field of freight car brake system diagnosis.


Introduction
The air brake system is a crucial component of train safety, as it ensures speed control in regular operation and in emergency situations.The effectiveness of this system has a significant influence on the forces acting on the train and therefore has a direct impact on driving safety. [1,2]As the construction of full-scale models for testing is costly, numerical simulations have become the accepted method for studying air brake systems.Commonly used empirical models include look-up table and fitting formula approaches, which are primarily used in practical applications that DOI: 10.1002/adts.202301257require a general understanding of air brake systems, such as signal design and stopping distance estimation. [3]6][7] However, dynamic models based on physical principles are essential for improving system performance and predicting detailed characteristics.10] Despite their practicality, the existing dynamic models are not able to develop robust algorithms to classify fault types due to discrepancies between simulated and real operating conditions. [11]igital twins (DTs), virtual replicas of physical systems, are used in various sectors such as manufacturing, transportation, and health care due to their ability to optimize performance. [12,13][16] For example, DTs have been used to model the condition of locomotive parking brakes to classify their operating states, [17] as well as for real-time monitoring of brake pad wear, which increases the safety and efficiency of vehicles. [18,19]DTs face the challenge of accurately representing physical systems.The integration of machine learning algorithms into DTs can bridge the gap between simulations and real-world scenarios, [20,21] thus improving the accuracy of DTs.Deep learning methods such as Deep Neural Network (DNN), Convolutional Neural Network (CNN), and Deep Belief Network (DBN) have been applied to various high-speed train anomaly detection tasks and have demonstrated their ability to manage complex, high-dimensional data.These methods, presented in refs.[22-25] and [26, 27] require large data sets and hyperparameter tuning, which makes real-time implementation difficult.Chong et al. [28] demonstrated the potential of ANN and CNN using a 1-year high-speed train dataset, but also pointed out the limitations in data-poor scenarios and emphasized the need for more adaptable solutions.In this study, a model-based DT approach for fault classification in freight car braking systems is presented and verified by experimental validation.Using a physical model of the brake system, optimized parameters, and likelihood ratio tests, the DT is continuously updated to reflect potential fault conditions corresponding to those of the actual system.The research contributes to the development and diagnostic analysis of freight car braking systems using DTs and demonstrates the synergy between sound physical knowledge and model-based DTs to The study is divided into six sections: Section 2 provides the theoretical foundation and presents the physical model-based DT of the freight car braking system, Section 3 validates this DT with experimental data, Section 4 outlines a novel DT-based algorithm, Section 5 applies this algorithm to experimental data, and Section 6 concludes the study.

Typical Modern Air Brake Systems
The operating mechanism of a standard air brake system for trains is shown in Figure 1b.The heart of this system is a compressor mounted on the locomotive, which has the task of generating compressed air.This air is then stored in a main reservoir.The system is designed so that the train driver can control the flow of compressed air via a control valve, which either feeds air from the main reservoir into the brake pipe or releases air from the brake pipe into the atmosphere.In addition, the individual wagons are equipped with auxiliary reservoirs in which the compressed air is stored.The distributor valves, as shown in Figure 1a, play a central role in regulating the connections between these auxiliary reservoirs, the brake pipes, the brake cylinders and the external environment.When the brakes are applied, the compressed air flows from the auxiliary reservoirs to the brake cylinders and exerts force on the brake shoes, which in turn press against the wheel treads via the brake pads.Conversely, when the brakes are released, the distributor valves ensure that the compressed air is released from the brake cylinders into the atmosphere, thereby reducing the pressure on the brake shoes.

Distributor Valve Working Principle
The distributor valve, often considered a critical component of the air brake system, acts as its heart.Its primary function is to control the distribution of compressed air from the brake pipe to the auxiliary and control air reservoirs.It also detects pressure changes in the brake pipe, making it easier to apply and release the brakes.The distribution valve is connected to the brake pipe via a branch line and also connects to other important elements such as the auxiliary reservoir, the brake cylinders, and the control reservoir can be seen in Figure 2. The distributor valve controls the flow of compressed air from the brake pipe by first filling the control reservoir via a cut-off valve and then the auxiliary reservoir via a check valve.When the pressure in the brake pipe drops because the driver operates the brake valve, a pressure differential is created that lifts the hollow stem in the assembly and opens the main valve check valve.This allows pressure from the auxiliary reservoir to enter the brake cylinder through a limiting device and apply the brakes.The main valve and this limiting device work together to limit the pressure in the brake cylinder to a maximum of 3.2 Bar.As the pressure builds up in the brake cylinder, it acts on the upper diaphragm of the main valve and causes it to move down together with the check valve until it reaches the lap position where the check valve of the main valve closes and prevents further increase of the pressure in the brake cylinder.

Distributor Valve Modeling
During the development of the DT model, the focus was on scalability and adaptability in different operational contexts, especially for different freight car braking systems.To achieve this, the model was designed with a modular architecture that allows for flexible adaptation to the unique dynamics and requirements of different applications.This modularity facilitates the precise tuning of parameters and subsystem configurations to accurately reflect the specific mechanical characteristics of each target system.However, adapting the model to new systems presents some challenges.One major hurdle is obtaining the detailed, systemspecific data required for accurate parameter calibration.In addition, the complexity of modeling unique mechanistic phenomena that may not be present in the original system design requires the creation of new sub model constructs.Despite these obstacles, the basic modular concept of our DT model creates a versatile framework that greatly enhances its applicability to a broad range of freight car braking systems and offers promising opportunities to improve predictive maintenance and operational efficiency.

Main and Check Valve Dynamics
The main valve's, which is shown in Figure 2, main function is to modulate the pressure of the brake cylinder in response to pressure changes in the brake pipe, thus facilitating pressurization for braking and subsequent release.The main valve consists of two diaphragms: a larger and a smaller one.The larger diaphragm is located at the lower end of the spindle and is exposed to the pressure in the brake pipe on the upper side and to the pressure in the control reservoir on the lower side.Conversely, the smaller diaphragm, which is located at the upper end of the spindle, is exposed to the pressure of the brake cylinder on its upper side and to atmospheric pressure on its lower side.A check valve, located at the upper end of the hollow spindle, regulates the connection between the auxiliary reservoir and the brake cylinder.This valve is sometimes referred to as a "three pressure valve" as it interacts with different pressure zones.
The dynamics of the main valve can be described by the following piece wise differential equation: Equations (1-6) show the dynamic interactions of the forces and pressures acting on the main valve and the check valve.These include in particular the forces due to the atmospheric pressure P atm , the brake cylinder pressure P bc (t), the brake pipe pressure P B (t) and the control pressure P C (t).In addition, there are the diaphragm areas A S and A L , which represent the smaller and larger diaphragm, respectively.The equations also contain the mass of the main valve M M , the damping coefficients C M and C cv as well as the spring stiffness K M and K cv .In addition, V ATL represents the amount of air flowing to the limiting device through the check valve and D cv is the diameter of the lower air passage of the check valve, ṁ(t) is the mass flow to the limiting device and OCV x M is the initial distance between the main and the check valves.The use of the differential notation and dX M (t) dt elegantly capture the acceleration and velocity of the main valve and thus provide a comprehensive overview of the dynamics of the valve under different pressure conditions.

Quick Service Valve Dynamics
The quick service valve has two different diaphragms: an upper and a lower one, as can be seen in Figure 2. The upper diaphragm, which is strategically positioned in the valve assembly, is pressurized on the upper side by the control reservoir, while its lower side is pressurized by the pressure in the brake pipe pressure.The lower diaphragm, on the other hand, which is located below the upper diaphragm, is exposed to the pressure of the brake pipe pressure on its underside during the braking process.The primary role of the quick service valve is to initiate a rapid reduction in brake pipe pressure.This is achieved by the pressure in the brake pipe pressure flowing abruptly into the expanding volume of the bulb at the start of the braking process.Such a mechanism is crucial for the rapid transfer of the pressure drop over the entire length of the train and improves the responsiveness and effectiveness of the braking system.This initial pressure drop generated by the quick service valve plays an important role in maintaining the operational efficiency and safety of the train's braking mechanism.The dynamics of the quick service valve can be described by the following piece-wise differential equation: P Brake Pipe (t) = max(P Brake Pipe (t), 3.2 × 1e5) (Ensuring minimum brake pipe pressure) ( 14) Equations (7-13) model the response of the valves under different conditions, which are divided into three different cases based on the displacement of the valves; 1) When the valve displacement X Q (t) is below the critical threshold XQ uo , the equation includes the interaction between the differential pressures-the control pressure P C (t) and the brake pipe pressure P B (t).It also takes into account the area of the upper diaphragm AQ up , the mass of the piston M Q and the dynamic properties determined by the damping coefficient CQ up and the spring stiffness KQ up .2) When X Q (t) enters the intermediate range-more precisely, between XQ uo and XQ maxopening -the dynamics of the valve continue to develop.The equation now integrates additional variables: the damping coefficient of the lower diaphragm CQ low and its corresponding spring stiffness KQ low .This shift in the parameters reflects the changed behavior of the valve when it transitions from a closed to a semi-open state and modulates the flow and pressure accordingly.3) When the valve reaches or exceeds the threshold value XQ maxopening , which means a fully open state.In this scenario, the equation simplifies and shows a static position with no further dynamic changes as the valve has reached its maximum operating capacity.
These displacement dynamics are complemented by the equations that control the dynamics of the air volume and pressure.The volume of air flowing to the bulb V ATB (t) is determined by the open position of the valve X Q open (t) and the valve diameter D Q valve down .The mass flow rate to the bulb ṁ(t) and the subsequent pressure in the bulb P Bulp (t) are calculated from this volume, the brake pipe pressure P Brake pipe (t), and the thermodynamic constants R and T. In addition, the brake pipe pressure P Brake Pipe (t) is adjusted to maintain a minimum threshold value that ensures the operational integrity and safety of the system.

Limiting Device and Diaphragm Dynamics
The limiting device consists of a single diaphragm, as shown in Figure 2.This diaphragm is designed so that its lower side is exposed to the pressure of the brake cylinder during braking, while the upper side is exposed to the combined pressure of a compression spring and the atmosphere.The main function of this limiting device is to regulate the maximum pressure in the brake cylinder to a precise value of 3.2 ± 0.1 bar.This regulation is independent of fluctuations in the pressure in the brake pipe and in the auxiliary reservoir.The limiting device plays a crucial role in ensuring constant and safe braking performance under different operating conditions.By limiting the pressure in the brake cylinders, it prevents over-braking and balances the braking force acting on the train.The operation of the limiting device and its effect on the overall braking dynamics can be described in a mathematical model as follows: Equation ( 15) models the acceleration of the diaphragm, (where X D (t) denotes the diaphragm displacement), which is defined by the pressure difference across the diaphragm, the diaphragm area A D , the mass M D , the damping coefficient C D and the spring stiffness K D .The pressures taken into account are the pressure in the brake cylinder P LD ud and the atmospheric pressure P bc .The velocity and the position X D (t) of the diaphragm are updated based on this acceleration and its previous values, with constraints to ensure that the displacement of the membrane remains within physical limits.In addition, V AAR(t) represents the amount of air flowing to the brake cylinder from the auxiliary reservoir through the limiting device and D dd is the diameter of the lower air passage of the limiting device.

Piston Dynamics in Brake Cylinder
The cylinder is equipped with the load spring, as can be seen in Figure 2. The piston chamber is hermetically isolated from the service brake chamber and is additionally separated by an atmospheric sluice.The piston dynamics can be mathematically described by the following equations: P bc (t) = max(P bc (t), 0) (Ensuring pressure does not go below zero) The piston's acceleration (where X P (t) is the piston's displacement) is determined by the pressure differential across the piston, the piston's area A P , mass M P , damping coefficient C P , and spring stiffness K P .The pressures involved are the brake cylinder pressure P ps1 and atmospheric pressure P ps2 .The velocity dX P (t) dt and position X P (t) of the piston are updated accordingly.The brake cylinder pressure update is modeled through a differential equation.The rate of change of brake cylinder pressure dP bc dt is calculated based on the mass flow rate ṁ, specific heat ratio , volume of the brake cylinder V, specific gas constant R, and temperature T. The updated brake cylinder pressure P bc (t) considers the piston's movement.

Experimental Validation of the Proposed Distributor Model
The use of the DT model in railway braking systems requires the integration of precise sensor systems and ensuring their maintenance to guarantee the accuracy of the data.It also requires advanced computing resources for real-time data processing and simulations.Challenges include integrating the model into current railway operations, which requires technical compatibility, operational adaptations, and specialized training for staff.
In this study, the DT model integrates real-time data via an M3 Speedgoat controller equipped with advanced processing capabilities (Intel 2GHz quad-core CPU, 4GB RAM, 64GB SSD) and tailored to the realistic simulation of railway braking systems.The DT achieves a data sampling rate of 100 Hz and uses four high-precision pressure sensors (P1A-55G-1-A-06-C-E), ensuring accurate brake pressure measurements with redundancy to protect against sensor failure.In this study, a 55-s interval was chosen for batch processing based on the computational resources of the Speedgoat controller, enabling the complex calculations of the algorithm without compromising the performance of the real-time system.We implement a real-time filter to exclude implausible data, e.g.brake pressures outside the physical Using the DT of the freight car, four primary operating states were simulated in this study: healthy braking system, a defective spring in the main valve, a defective spring in the check valve, and a defective spring in the limiting device.The simulations were performed under the premise that the total thermal energy of the gas remains constant during compression, in accordance with the ISO 6358 standard.The specific heat ratio for air () was set to 1.4, whereby the gas constant (R), the temperature (T), and the ambient pressure (P atm ) were each determined at 278 J kg -1 K -1 , 25 °C, and 100 kPa.In order to accurately reflect the complexity of the braking system, we have precisely defined the parameters for the main valve, the limiting device, quick service valve, and brake cylinder.These parameters are detailed in Table 1.
Figure 3 serves as a comprehensive demonstration of the versatility of the DT model and shows three different use cases that encompass a range of braking scenarios.These scenarios include normal braking with a healthy system, gradual brake and release modulation with a healthy system, and a fault condition within the braking system, as shown in Figure 3a-d.This diverse representation is critical for validating the ability of the DT model to accurately simulate the dynamic behavior of the railway braking system under different operating and fault conditions.In the first case, shown in Figure 3a,b, the experiment was carried out in several steps.First, the brakes were applied, which led to a rapid pressure drop in the brake pipe pressure (as shown by the red curve in Figure 3a).This pressure drop caused a differential across the large diaphragm of the main valve assembly, which lifted the hollow stem of the main valve and opened the check valve.As a result, the pressure from the auxiliary reservoir entered the brake cylinder via the limiting device and applied the brakes (as indicated by the blue and yellow curves in Figure 3a).The increase in pressure in the brake cylinder exerted a force on the upper diaphragm of the main valve, causing the main valve and check valve to move downwards until they reached the lap position, limiting the pressure increase in the brake cylinder to a maximum of 3.2 bar (see Figure 3a,b).In this state, the check valve closed and prevented a further pressure increase in the brake cylinder.The pressure in the brake pipe was then maintained (lapped) for 18 s, with the pressure in the brake cylinder remaining constant.When the brake was released (indicated by an increase in pressure in the brake pipe, red curve in Figure 3a,b, the differential pressure across the large diaphragm of the main valve neutralized, allowing the hollow stem assembly to return to its original position and the pressure in the brake cylinder to escape to atmosphere, as shown by the blue and yellow curves in Figure 3a.In this study, the parameters K cv , C cv , K mainvalve and C mainvalve were prioritized based on their critical role in the dynamics of the braking system and updated and optimized using the Nelder-Mead simplex algorithm.These parameters were selected based on maintenance data and operational analysis which shows that the main valve and check valve springs fail most frequently and have a significant impact on braking performance.Using the optimization process, these parameters were used to fit the DT model to the experimental results and classify the condition of the braking system, as explained in Sections 4 and 5.This optimization is evident in the comparison between the DT model (blue curve) and the experimental results (yellow curve), especially with regard to the pressure in the cylinder chamber before and after the optimization of the model parameters, as shown in Figure 3a,b, which show a close match in both the amplitude and pattern of pressure changes.Any gradual reduction of the pressure in the brake pipe reproduces this phenomenon and modulates the pressure in the brake cylinder, as can be seen in Figure 3c.The validity of the DT model was confirmed in three failure scenarios.Figure 3d shows the agreement of the DT model results with the experimental results in a scenario with a faulty spring in the main valve assembly, which restricted the pressure flow and limited the pressure in the brake cylinder to 2.2 bar, which is crucial for the safety of the train.These results underline the ability of the DT model to accurately replicate the behavior of brakes in the real world and confirm the accuracy and reliability of the underlying mathematical models.

The DT Algorithm
The proposed algorithm is integrated into the freight car DT.It consists of four steps, illustrated in Figure 4.
Step 1: The pressure in the brake pipe and brake cylinder (indicated by P Brake Pipe and P bc ) were routinely measured in the freight cars tested.In this study, the internal latent physical variables x = [K cv , C cv , K mainvalve , C mainvalve ] are estimated by minimizing the objective function, which quantifies the deviation between the simulated output of the model and the measured signal.The objective function to be minimized is given by: where model denotes the simulated brake cylinder pressure and X Measured represents the measured cylinder pressure.
Step 2: The optimization process utilizes the Nelder-Mead simplex algorithm, which employs a simplex of n + 1 points for n-dimensional vectors x.The algorithm, as described in Lagarias et al., [29] proceeds as follows: 1. Initialization: Let x(i) denote the points in the current simplex, where i = 1, … , n + 1.The simplex is initialized around the initial guess x 0 .
2. Ordering: Order the simplex points from lowest to highest function values, f (x(1)), … , f (x(n + 1)). 3. Reflection: Reflect the worst point x(n + 1) across the centroid m of the other n points: Compute f (r) and proceed based on the following conditions: 4. Acceptance Criteria: , accept r and terminate this iteration.
), compute the expansion point s: Compute f (s).If f (s) < f (r), accept s and terminate the iteration; otherwise, accept r.
, perform a contraction between m and either x(n + 1) or r, based on the lower objective 5. Update Simplex: The next iteration uses the simplex formed by x(1), v(2), … , v(n + 1).
This iterative process adjusts the simplex to explore the parameter space, aiming to find the point that minimizes the objective function f (x), leading to the optimal parameter set x optimal .
Step 3: The classification algorithm begins by dividing the empirical data into subsets, each of which corresponds to a partic-ular class or type of fault.If there are a total number of Classes, the data sets are subdivided accordingly.For each class (or data set) j, the algorithm initializes two arrays to store the means and covariances for each class and a matrix X to store the calculated probabilities.The mean vector  j and the covariance matrix Σ j for each class j are calculated using the empirical data for that class.These statistical parameters are crucial for the calculation of the multivariate normal probability densities.They serve as a representation of the central tendency and dispersion of the multivariate data within each class.The calculation is performed as follows: where D i,j is the i-th sample in the data set j and N j is the number of samples in the class j.These parameters are then used in the calculation of the multivariate normal probability density for each sample with respect to each class, as described in the following section on probabilistic classification.Then iterates over each sample and each class to compute the multivariate normal probability density for each sample, relative to each class mean and covariance.This is represented by the matrix X, where the element X ct,i,j is computed as: Here, D i,ct is the ct-th sample in dataset i,  j , and Σ j are the mean and covariance of class j, respectively, and k is the number of dimensions in the data.
Step 4: The class assignment for each sample is determined by selecting the class with the highest probability.For each sample ct, the assigned class C ct is the one for which the probability is maximum.This can be mathematically represented as: where C ct is the class assigned to the ct-th sample, X ct,i,j is the probability of the ct-th sample belonging to class j, and numClasses is the total number of classes.

Fault Diagnosis
Figure 5 shows an example of the visualization of the iterative optimization process of a healthy braking system, which is crucial for the health monitoring system for braking systems using the proposed DT. Figure 5 illustrates the convergence of the model parameters K cv , C cv , K mainvalve , and C mainvalve over 37 iterations of the optimization algorithm.As can be seen in Figure 5, the parameter K cv starts with a maximum value of 5195 N m -1 , indicating an initial estimate that is far from optimal, and shows a descending trend before stabilizing at a value of 2500 N m -1 , re-flecting the optimal value for a healthy braking system.The parameter C cv , which starts at 150 Ns m -1 , also shows a clear decrease, indicating a corresponding refinement of the diagnostic accuracy over the course of the algorithm.In parallel, the parameter K mainvalve is initially observed at 1785 N m -1 , which gradually increases and then stabilizes in the range of 1930 N m -1 , indicating that the algorithm is adapting to the optimal conditions for the state of the brakes.The parameter C mainvalve follows a similar stabilization pattern, further supporting the reliability of the optimization process.The most convincing aspect of the optimization is captured by the objective function f (x).Starting from 61107.8, f (x) is drastically reduced to 0.04, illustrating the algorithm's ability to identify and narrow down the most favorable system state for a healthy braking system.The ability of the iterative process to update the model parameters and converge to a representation indicative of a healthy braking system is a testament to the potential of the proposed DT architecture for real-time monitoring and preventive maintenance strategies.The optimization results for all faults showed similar results in converging the parameters to their expected values in each fault condition.
The new algorithm described in Section 4 was tested and validated on 200 test examples, 50 from each of the conditions mentioned in Section 3. The results of the tested algorithms are shown in Figure 6.Each boxplot in Figure 6a-c shows considerable variability in response to four working conditions.Figure 6a shows remarkable consistency in the main valve spring constants across all conditions, with a slight deviation observed when a fault occurred in the main valve spring.The low standard deviations across all conditions, including the faulty conditions, indicate stable and predictable performance, a desirable characteristic in system critical components.Figure 6b shows significant variations between the different operating conditions.A notable increase in the mean value of the spring constant was observed in the condition where the limiting device spring was faulty, while it decreased significantly in the condition with the faulty check valve spring.The high standard deviation indicates a wide range of operating behavior, possibly reflecting the response of the system to the specific fault.Figure 6c shows that the mean limiting device spring constant varies considerably, with the much higher value at the limiting device spring fault being the most noticeable.This indicates that the limiting device is more sensitive to its operating fault condition, which requires careful monitoring and adjustment.Based on these findings, the proposed algorithm is able to classify the healthy state and the specific fault states of the components of the DT.The algorithm showed a high level of accuracy, achieving a classification accuracy of 97.5%, as shown in Figure 6d.This high level of accuracy demonstrates the DT ability to improve diagnosis by incorporating physical knowledge of the system.

Summary and Conclusions
This study presents a novel DT algorithm developed for the diagnosis of faults in freight car braking systems.The algorithm is a four step process that includes the estimation of internal DT variables, the application of the Nedler-Mead simplex algorithm, and fault classification and prediction.The algorithm was rigorously tested on a data set of 200 test examples and showed an impressive accuracy rate of 97.5%.A special feature of the DT algorithm is the inclusion of physical knowledge of the braking system, which facilitates accurate diagnosis.The approach is based on model-based fault detection and diagnosis, which relies mainly on standard measurements of the pressure in the brake pipe.This methodology uses a physically derived brake model that offers broad applicability under different operating conditions and ensures transferability to different freight car brake systems.The symptoms identified by DT are generally easy to interpret, which increases its practicality.The model-based DT approach described in this study represents a significant advance in engineering performance and goes beyond traditional engineering design paradigms.However, it is important to know the conditions of applicability and the possible limitations of the DT algorithm we propose.The effectiveness of this approach depends on the quality of the data collected from the braking systems as well as on the accurate representation of the physical system dynamics within the DT.While our algorithm shows broad applicability under different operating conditions, its performance may fluctuate in scenarios where the braking system undergoes changes that were not accounted for in the original model or under extreme conditions that were not part of the dataset used for testing.In addition, the successful application of this method requires a fundamental understanding of the underlying physical principles of freight car braking systems and the ability to accurately interpret the DT output.These conditions emphasise the importance of continuous model validation and possible recalibration to adapt to new data or system changes to ensure that the DT algorithm remains a robust tool for fault diagnosis.This DT algorithm not only impacts train operations through improved diagnostic capabilities, but also revolutionizes maintenance activities.By effectively detecting and correcting maintenance faults, the DT algorithm contributes significantly to improving maintenance efficiency.The overarching benefit of the DT algorithm is its profound impact on customer experience and operational costs.By providing real-time performance data to an adaptive system encapsulated in a digital layer, DT is a testament to the value of performance-driven engineering.This study illustrates the transformative potential of DTs in industrial applications.The DT algorithm developed here not only ensures the safety and efficiency of freight car braking systems, but also paves the way for smarter maintenance and operational strategies that ultimately increase customer satisfaction and reduce operating costs.

Figure 1 .
Figure 1.Freight car air brake system.a) location of freight car braking system, b) schematic of a typical freight car air brake system.

Figure 2 .
Figure 2. Schematic diagram of the distributor valve.

Figure 3 .
Figure 3. Model results show the response of the pressure in the cylinder chamber for different brake operation modes compared with experimental results.a) Prior to model parameters optimization, b) after model parameters optimization, c) step braking model parameters optimization, and d) main valve spring fault.

Figure 4 . 29 )
Figure 4.The new suggested algorithm for machine fault diagnosis using a DT described in Section 4.

Figure 5 .
Figure 5.The new suggested algorithm for machine fault diagnosis using a DT described in Section 4.

Figure 6 .
Figure 6.Results of the new suggested algorithm.a) Variation of spring Constant of the main valve, b) variation of spring Constant of the check Valve, c) variation of spring constant of the limiting device, and d) the summarized accuracies of (a-c).

Table 1 .
Parameters for main valve, limiting device, quick service salve and brake cylinder.