## 2 Materials and Methods
### Vehicles
#### 2.2.4 Train Simulator (PBLM-Rail)  
The simulation is based on several sub-models that exchange information during the simulation. The model for vehicles is physical-based, and it has been developed in detail in previous papers [19]. It applies a scalar equation to calculate the acceleration of the train (DMU, EMU, DHEMU, or locomotives with wagons or coaches). The equation expresses the equilibrium between external and internal forces:  
\[\left(m+\frac{J}{R^{2}}\right)a=\sum_{k=1}^{n}\left(F_{k}\right) \tag{1}\]  
\[m(1+ERM)a=\sum_{k=1}^{n}\left(F_{tk}\right)-\sum_{k=1}^{n}\left(F_{rk}\right)- F_{c}-F_{g}-F_{s}-F_{b} \tag{2}\]  
where  
\(n\): overall number of effects being considered.  
\(m\): mass of the train  
\(a\): acceleration of the train  
\(J\): moment of inertia of the different rotating masses (wheels, motor rotors...) in the transmission, which causes an apparent increase of mass.  
\(R\): wheel radius.  
\(F_{tk}\): propulsive force, tractive effort of the k locomotive, DMU, EMU,...  
\(F_{rk}\): resistance forces due to the element r of the train, locomotive, coach, generator...  
\(u\): number of units in the train  
\(F_{c}\): curve resistance  
\(F_{g}\): resistance due to the slope of the track (positive or negative)  
\(F_{s}\): starting resistance  
\(F_{b}\): resistance due to braking, in our case dynamic braking [20] and pneumatic braking [21].  
In this model, ERM is the Equivalent Rotational Mass, which ranges from 0 to 0.30, depending on the type of vehicle. Other values can be found in [22; 23].  
Resistance forces are the sum of all forces acting on the railway unit at a given time or place. These resistance forces can be examined in detail in the literature by different models [20; 23], but they are not the main objective of this paper. To approximate rail vehicle resistance in PBLM-Rail, this paper applies the equation:  
\[F_{r}(v)=A+Bv+Cv^{2} \tag{3}\]  
where \(v\) is the speed of the vehicle (m/s), and \(A\) (N), \(B\) (Ns/m), and \(C\) (Ns\({}^{2}\)/m\({}^{2}\)) are regression coefficients obtained by fitting the run-down test of passenger and freight units to the previous equation.  
This paper uses different data from manufacturers such as Bombardier, Alstom, Patentes Talgo, Stadler Rail, Siemens, Electro-Motive Division (EMD), Mitsubishi, and railway operators (SNCF, DB, or SBB).  
With respect to grade resistance with slope \(\beta\), the force is  
\[F_{g}=mgsin(\beta) \tag{4}\]With respect to curve resistance, literature approaches use different empirical equations for different curve radii. For example, in reference [24]:  
\[F_{c}=0.001 mg\ *\ 198/r_{c} \tag{5}\]  
where \(r_{c}\) is the curve radius.  
With respect to starting force \(F_{s}\), there are few studies on this topic. An accurate approximation of the main interest for haulage (and specifically for freight) is the approach proposed in reference [25]:  
\[F_{s}(v)=\left\{\begin{aligned} 3F_{r}(10);\ v=0\ \text{km}/\text{h} \\ 2F_{r}(10)(1-v)+F_{r}(1);\ v\in(0,\ 1]\ \text{km}/\text{h}\\ F_{r}(10);\ v\in(1,\ 10]\ \text{km}/\text{h}\end{aligned}\right. \tag{6}\]  
As seen in Equation (6), the approach is that the value of F\({}_{\text{S}}\) is related to F\({}_{\text{r}}\) in Equation (3).