*2.4. Economic Model*

The annual cost of energy is used to compare the different technical solutions proposed. Therefore, it is necessary to estimate the present value of the total cost, which includes: the cost of capital, the present value of the operating costs and the present value of the replacement cost of the energy storage system [45]. The ACOE is mathematically expressed as:

$$\text{ACOE} = \text{CRF} \cdot \text{C}\_{TOT} \tag{15}$$

where *CRF* is the capital recovery factor converting a present value into a stream of equal annual payments over a specified lifetime *N*, at a specified interest rate *r*, and *CTOT* is the present value of the total cost. The capital recovery factor is computed by using the following equation:

$$\text{CRF} = \frac{r(1+r)^N}{\left(1+r\right)^N - 1} \tag{16}$$

### 2.4.1. Costs of an Electrified Railway Line

Measuring the performance of an electrified railway is particularly complex as it involves a service that requires rolling stock, tracks, safety and signaling systems, stops or stations, and a variety of personnel types. Another factor that affects transport, if it is public, is governmen<sup>t</sup> intervention to subsidize costs [49]. In the present study, the cost of capital associated with fixed electrical systems has been estimated based on the main components that characterize this scenario:

$$\mathbf{C}\_{I}^{ELE} = \mathbf{C}\_{TPS} + \mathbf{C}\_{CAT} + \mathbf{C}\_{MSC} \tag{17}$$

where *CELE I* is the capital cost of the electrification of the railway line, *CTPS* is the cost associated with the traction power substation, *CCAT* is the cost associated with the installation of the catenary and *CMSC* represents other costs related to the electrification intervention such as track lowering in correspondence of tunnels already present in the route and raising of the overpasses. The present value of operation and maintenance (O&M) costs associated with the electrified railway line is calculated as follows:

$$C\_{O\&M}^{ELE} = \sum\_{n=1}^{N} \frac{C\_n^{ELE}}{(1+r)^n} \tag{18}$$

where *CELE n* is the annual operations cost of year *n* including both fixed and variable costs:

$$\mathbf{C}\_{n}^{ELE} = \mathbf{C}\_{t} \mathbf{N}\_{d} \mathbf{E}\_{d}^{TPS} + \mathbf{C}\_{n}^{TPS} + \mathbf{C}\_{n}^{CAT} \tag{19}$$

where *ETPS d* is the energy supplied by the traction power substation in a day, during *Nd* days within a year, *Ce* the average cost of electricity, *CTPS n* and *CCAT n* represent the average of the annual costs associated with the maintenance of the traction power substation and the catenary, respectively, estimated from 1% to 3% per year of the investment cost, referring to the date of commissioning of the equipment [50]. Finally, the annual cost of energy is calculated by using the following equation:

$$ACOE\_{ELE} = \text{CRF} \cdot \left( \mathbf{C}\_{I}^{ELE} + \mathbf{C}\_{O \& M}^{ELE} \right) \tag{20}$$

2.4.2. Costs of Associated with the Use of Trains Equipped with On-Board ESS

The costs taken into account are the capital cost of the on-board ESS, the annual operation costs of the on-board ESS, operation and maintenance (O&M) costs of the on-board ESS and the replacement cost of the on board ESS.

The use of trains equipped with ESS might reduce capital costs as there is no catenary, but the construction of new specialized trains could significantly increase the total costs of realization. Moreover, depending on how the sizing of the energy storage system is carried out, it may be necessary to install a charging system with an installed power comparable to that of a traction power substation, thus a ffecting capital costs. Furthermore, it is necessary to respect the limits of the state of charge within which the ESS must deliver and absorb energy, even with high discharge current peaks provided they occur for short periods of time, resulting in a useful life of 10÷15 years [51]. However, is needed an expertise in train operation and real work cycle of storage system to know if this useful life is respected.

Consequently, the cost assessment must include the future replacement of the energy storage system following life expectancy and battery life reported in data sheet (not considering battery disposal).

In cost assessment is not taken into account the cost due to the buy of new trains, but is assumed that the cost of the on board ESS is the di fference cost between traditional electric trains and new trains equipped with on board ESS. So, the following analysis is relevant on the hypothesis of complete renewal of the rolling stock, as diesel trains.

The cost of capital of the equipment of the storage systems *C*ESS *I*, has been estimated as follows:

$$\mathbf{C}\_{I}^{\text{ESS}} = \mathbf{C}\_{P}\mathbf{P}\_{\text{ESS}} + \mathbf{C}\_{E}\mathbf{E}\_{\text{ESS}} + \mathbf{C}\_{\text{FC}} \tag{21}$$

where *CP* [€/kW] and *CE* [€/kWh] are the ESS specific costs, *P*ESS and *E*ESS are the power and energy capacities and *CFC* are the ESS fixed costs associated, for example, with the installation of a recharging system. The present value of operation and maintenance costs associated with the use of trains equipped with on-board ESS *C*ESS *O*&*M*, is calculated by:

$$\mathcal{C}\_{\text{O\&M}}^{\text{ESS}} = \sum\_{n=1}^{N} \frac{\mathcal{C}\_n^{\text{ESS}}}{(1+r)^n} \tag{22}$$

where *C*ESS *n* is the annual operations cost of year *n* including both fixed and variable costs, and is computed by using the following equation:

$$\mathcal{C}\_{\text{H}}^{\text{ESS}} = \mathcal{C}\_{f} P\_{\text{ESS}} + \mathcal{C}\_{v} N\_{d} \, E\_{d}^{\text{ESS}} + N\_{d} \, \frac{E\_{d}^{\text{ESS}}}{\eta\_{ch}} \mathcal{C}\_{ch} \tag{23}$$

where *Cf* [€/kW-yr] represents the specific operating costs, *Cv* [€/kWh] the variable operating costs, *Cch* [€/kWh] the cost of recharging ESS, *Nd* the number of days the ESS is active within one year, *P*ESS [kW] and *EESS d* [kWh] the nominal power and energy supplied by the ESS, and finally η*ch* is the charge e fficiency. The replacement cost of the ESS is expressed in the following equation:

$$\mathcal{C}\_{R}^{\text{ESS}} = \mathbb{C}\_{FR} \left[ \left( 1 + r \right)^{-L\_R} + \left( 1 + r \right)^{-2L\_R} + \dots \right] \tag{24}$$

where *CFR* represents the future value of replacement cost and *LR* is the ESS lifetime. Finally, the annual cost of energy is calculated by using the following equation:

$$ACOE\_{\rm ESS} = \mathcal{CRF} \cdot \left( \mathcal{C}\_{I}^{\rm ESS} + \mathcal{C}\_{O\&M}^{\rm ESS} + \mathcal{C}\_{R}^{\rm ESS} \right) \tag{25}$$
