Train Performance Simulation (TPS) is the process of determining a train’s position, speed, power consumption, and other relevant factors over time as it traverses a specific track section. To effectively simulate a train’s performance, it is crucial to accurately compute acceleration and deceleration forces, considering factors such as traction, braking forces, and various resistance forces related to the track’s geometry.

Precise evaluation of train performance necessitates the calculation of acceleration and deceleration forces by considering traction and braking force curves as functions of speed. Additionally, it requires an understanding of rolling resistance as a function of speed, along with gradient and curvature resistance. By using TPS, the status of the train’s mechanical motion, such as position, speed, and traction, can be determined at each moment. Consequently, TPS calculates the train’s power consumption based on this information.

Typically, TPS takes both track information and train information as inputs. Utilizing this data, it simulates the train’s motion along the track, allowing the train’s performance to be analyzed as a function of speed, time, and distance. This comprehensive approach enables a more accurate assessment of various factors affecting train performance, ultimately enhancing the efficiency and safety of railway operations.

The force exerted by a locomotive to pull a wagon while simultaneously propelling itself forward is known as the tensile force. As a train travels along its route, the change in its speed is influenced by the relationship between the pulling force and the rolling resistance. When the tensile force is more prominent, the train accelerates. Conversely, when the resistance is larger, the train decelerates. If the tensile force and resistance are equal, the train maintains its current speed. Tensile forces can be broadly categorized into the following three groups:

· Tractive force: This refers to the tensile force constrained by the primary motor current during acceleration. A more considerable starting current enables a more significant tensile force to be exerted.

· Characteristic tensile force: This term encompasses the tensile force restricted by the primary motor or diesel engine’s characteristics. As the speed of the train increases, the characteristic tensile force decreases correspondingly.

Adhesive tensile force: This force emerges when the characteristic tensile force is applied to the wheel. A friction force is subsequently generated between the wheel’s circumference and the rail surface, preventing rotation. This friction force is subject to limitations, with the maximum tensile force permitted before rotation occurring referred to as the adhesion tensile force.

As illustrated in Figure 1, the tensile force of a locomotive exhibits distinct characteristics. As the train’s speed increases, the tensile force is denoted by the starting tensile force, adhesive tensile force, and characteristic tensile force, respectively. The traction force displays similar characteristics that are contingent

on the train’s speed. The speed ranges for the constant torque, constant power, and characteristic regions correspond to each of these tensile force types.

The train’s acceleration is influenced by the traction power of the motor and the resistance it encounters. Equation (1) demonstrates that the acceleration is determined by dividing the effective tractive force by the inertial weight.

a = F e f f m d y n (1)

where,

F_eff : The effective tractive force [kN];

m_dyn: The dynamic mass [ton].

The effective tractive effort can be calculated by subtracting the train resistance from the electric motor tractive effort, as depicted in Equation (2).

F e f f = F m t f − R (2)

R = R r u n + R c u r + R g r u (3)

where:

F_mtf : The electric motor traction force [kN];

R: The train resistance [kN];

R_run: Running resistance [kN];

R_cur: Curvature resistance [kN];

R_gra: Gradient resistance [kN].

The train resistance (R) is composed of three components: curvature resistance, running resistance, and gradient resistance. Equation (3) presents the summation of these resistances.

The force that decelerates a moving train during braking is referred to as the braking force. When braking, the train encounters resistance, as depicted in Equation (4). If the gradient is uphill, the gradient resistance is added to the braking force, acting in the direction of further reducing the braking force when the gradient is downhill.

B = B m r t f + R r u n + R c u r + R g r u (4)

where:

B_mtf : The braking force of the vehicle [kN];

R_run: Running resistance [kN];

R_cur: Curvature resistance [kN];

R_gra: Gradient resistance [kN].

The braking force characteristics of the vehicle are illustrated in Figure 2.

To calculate the actual motion of the vehicle, either Equation (2) or Equation (4) is applied based on the control state of the train.

In the motor’s traction and braking force characteristic curve during vehicle operation, there exists a relationship between voltage change and the corresponding speed. This relationship can be obtained through Equations (5) and (6).

F e f f = { V d c × v v c o n s t ( v < v c o n s t ) V ( v ≥ v c o n s t ) (5)

I m o t o r = F × v V m o t o r × cos θ × η = P V m o t o r × cos θ × η (6)

where:

V_motor: The motor voltage [V];

I_motor: The motor current [ton];

V_dc: The motor rated voltage [V];

v: The vehicle speed;

v_const: The reference speed for traction or braking;

F: The traction or braking force;

P: The vehicle power;

η: The traction or regeneration efficiency.

Figure 3 illustrates the time-varying speed change during reverse travel, calculated using the proposed model. In contrast, Figure 4 displays the measured data of the time-varying speed curve [8] . Notably, the vehicle maintains a consistent speed of up to 35 km/h. However, beyond this threshold, the acceleration decreases, allowing the speed to reach its maximum.

Figure 5 presents the model-calculated time-dependent curves for power usage and regenerative power, with the braking force region highlighted in red. In Figure 6, we display the measured data of the time-dependent braking force curve. Notably, the simulated braking force curve aligns with the observed electric braking force. Furthermore, it is evident that the electric braking force decreases as the air braking force, necessary for braking, gradually increases.

Through a comparison of the proposed model’s calculations of the vehicle’s speed during reverse and regenerative power during braking characteristics with the measured vehicle data, we observe that the model accurately simulates the actual characteristics of vehicle motion and power.

The proposed model was implemented for the Seoul Metro Line 3 to conduct TPS (Train Performance Simulation), and the obtained results were compared with the measured data [9] . Table 1 presents the vehicle specifications of the electric vehicles used as input data for TPS.

To determine the inertial mass of a train formation, the individual inertial masses of each vehicle within the formation are summed. The inertial mass of a vehicle is calculated as the sum of its curb mass, inertia factor, and passenger load. The inertia coefficient is applied to the curb mass and varies depending on whether the vehicle is an M car or a T car. For instance, let’s consider a train formation consisting of 5M5T cars, with curb weights of 39 tons for M cars and 29 tons for T cars. The inertia coefficients are 14% and 5%, respectively, and the passenger load is 20 tons per car. In this case, the inertial mass of the first formation can be calculated as follows.

Table 2 presents the speeds that delineate the constant torque and constant power regions, representing the vehicle’s acceleration and deceleration characteristics.

The constant power area spans from 35 km/h to 65 km/h for towing and 60 km/h to 75 km/h for braking.

Regarding the TPS input data, the locations of the stations are specified in Table 3.

A train operation simulation was conducted using vehicle and track data, simulating a 10-car electric train operating between Gupabal and Seoul National University of Education (Seoul University of Education). The simulation results are depicted in Figure 7.

Regarding the consumption and regeneration power of the electric vehicles on Seoul Metro Line 3, the one-way section measurement results exhibited a reverse error rate ranging from 4.03% to 8.26%, and a regeneration error rate ranging from 2.30% to 5.74%. Comparing the error rates within round-trip intervals, it was observed that the reverse error rate differed by 6.19%, while the regeneration error rate differed by 3.97%. These findings indicate that the simulation results closely estimate the actual data.

When comparing the simulation results with the actual data, a natural correspondence is observed due to the equations of motion. The peak values of regression and recovery are nearly symmetrical in the actual data. However, the peak value of power during braking appears slightly larger than the peak value during reverse. Multiplying the peak value of braking by the ratio of electric braking to air braking yields a similar value to the actual data. This characteristic is applied in actual vehicle design. Hence, it is evident that considering traction/braking efficiency and accounting for the electric braking rate would enhance the accuracy

of power values calculated through simulation for traction and braking.