*3.2. Numerical Results*

Several simulations are carried out, imposing different upper and lower bounds of the input variables, for the case study described in Section 3.1 to highlight the strengths and weaknesses of the methodology implemented into the software. 20,000 iterations and 25 particles are imposed by running the proposed algorithm on a device with an Intel®Core ™ i7 processor (2.20 GHz, 64 bit), 16 GB of RAM and MATLAB™ R2019b. The economic inputs of ESS based on Li-ion battery, detailed in [23], are shown in Table 8.


**Table 8.** Economic inputs.

For the cost coefficient of electricity (Cch) a flat rate of 0.015 €/kWh has been considered. It is a realistic value compared to the tariff regime used in Italy for railways.

The variables managed by the program are: position, nominal power and nominal energy; the position value is chosen from those included between the beginning and the end of the line, the others are parameterized as shown in Table 9.

**Table 9.** Input parameters of the optimization algorithm.


The solution found by the optimization:


The solution provided allows to have the maximum possible profit and a relatively short payback period. Furthermore, making a comparison between the case without ESS and with ESS, less energy is drawn from the primary network and also a lower impact on the electrical substations, which are, especially those located in the middle of the line, less stressed, as shown in Figure 8b. Specifically, being able to count on the possibility of trains to reuse braking energy decreases the energy absorbed by the primary network by about 34% and the installation of an ESS along the line allows for about a further 8% less energy drawn. The electrical substations, on the other hand, thanks to the recovery of braking energy, deliver on average 30% less power (with peaks of over 40% for the substations located at the limits of the railway line), which is further lowered by about 5% in average (with peaks over 10% for the mid-line substations) thanks to the presence of ESS.

Moreover, it should be noted that through the ESS the electrical substations and the traction line are stressed less, with consequent benefits from the point of view of maintenance, aging and replacement of materials.

Even if it is not one of the specific objectives of the model, the adoption of an ESS along the line has the further benefit of reducing the system's energy losses (31 MWh against 32.9 MWh in a standard working day).

These aspects allow considerable savings in the long period and increase the advantages that can be obtained after installation: these are not taken into account by energy savings, but are estimated in the economic model on which PSO is based, so the solution provided allows to have greater overall benefits than looking only at the energy aspect.

**Figure 8.** Graphic software output: (**a**) Total energy withdrawn from the primary network in a working day; (**b**) Power supplied by each substation present on the line.

The operation of the line (timetable) is a fundamental role in the choice of the installation to be carried out. Using the same network and route characteristics of previous simulation, but with another timetable (only "soft" periods), it possible to appreciate that a much higher percentage of energy saving, taken into account ESS, is reached (Figure 9).

**Figure 9.** Graphic software output: (**a**) Total energy withdrawn from the primary network in a working day; (**b**) Power supplied by each substation present on the line.

The electrical substations load is more decreased, with benefits from the point of view of maintenance and replacement, as previously mentioned, which are not directly visible in energy consumption. Being able to simulate multiple service hours, it can be seen how this characteristic of the proposed model allows for a more accurate and realistic solution.

To avoid unusable solutions and excessively long and expensive simulation times, it is advisable to choose numerical ranges that have real ESS sizes and that are not too wide.

A further strength of the software is the possibility of providing more ESSs on the line: in fact, it is possible to impose a number of accumulators greater than 1 and to obtain the optimal solution from the software with that specific configuration. A simulation performed by predicting 2 ESSs in line, with the same ranges of decision variables predicted in Table 8, led to the follow solution:

• NPV: 383,451.71 €

• Payback period: 3.6648 years


The proposed solution involves a lower NPV than the case with only one ESS installed, but looking at the graphic outputs of the software (Figure 10), comparing the case without ESSs with that with two ESSs, an energy saving of about 14% is noted (compared to 8% in the previous case). The electrical substations deliver about 9% less power on average (before it was 5%), with peaks over 15% for the mid-line substations when 2 ESSs are installed according to the proposed method.

**Figure 10.** Graphic software output with 2 ESSs on the route: (**a**) Total energy withdrawn from the primary network in a working day; (**b**) Power supplied by each substation present on the line.

Also, from the point of view of the energy losses of the system, the solution proposed with two ESSs installed results in lower losses compared to the previous case with only one ESS installed (29.3 MWh against 31 MWh in a standard working day).

Despite a lower economic return from the installation, i.e. a slightly lower but still positive NPV, it is possible to lighten the electrical substations and draw even less energy from the network. The user who manages the software can evaluate the benefits of this solution through the graphic output, that allows to have all the information, both technical and economic, of the installation to be carried out.
