*2.2. DC Feeder System*

Conventional substations are represented by ideal DC voltage sources, series resistance and series diode if the substations are not reversible [28]. The contact wire is modelled as a set of electric resistances that change their value according to the vehicle position. If *x*(*t*) is the train position at the time *t*, the value of the resistance upstream *Ra* and downstream *Rb* to the vehicle towards a generic node of the railway feeding system (conventional substation or another train) are calculated by:

$$\begin{cases} \begin{array}{c} R\_{\mathfrak{a}}(t) = \rho \cdot \mathbf{x}(t) \\ R\_{\mathfrak{b}}(t) = \rho \cdot \left[ d - \mathbf{x}(t) \right] \end{array} \tag{6} $$

where *Ra* and *Rb* are expressed in [Ω], ρ [Ω/km] represents the resistive coefficient, *d* [km] is the distance between the two nodes (upstream and downstream the train). In order to improve the train electric model, describing the receptivity of the network under regenerative braking conditions, a small capacitance is connected in parallel to current source that represents the vehicle [46]. In Figure 2 it is shown the electric model of the overall railway system, one side supplied contact line.

**Figure 2.** Electric model of the one side DC supplied contact line.
