*3.2. ESS Sizing*

Given a commercial battery cell characterized by a nominal cell voltage *UCELLB* the number of cells to be connected in the series *NSEB* to obtain a given rated voltage of the battery pack, *UB* is obtained by:

$$N\_{SE\_B} = \frac{\mathcal{U}\_B}{\mathcal{U}\_{CELL\_B}}\tag{26}$$

It is highlighted that it represents a simplified model for the preliminary study developed in this paper, but in the case of a cell pack, balancing and operating safety should be added. The number of branches to be connected in parallel *NPARB* is determined as the maximum number between *NPPARB* , which represents the number of branches in parallel to satisfy the power requirement to be delivered in traction *Pmax B* , and *NEPARB* which is the one necessary to satisfy the energy required by the battery, *EB*. In order to increase battery lifetime, the variations in the state of charge are limited between a *SOCmin* of 0.2 ÷ 0.3 and a *SOCmax* of 0.8 ÷ 0.95. This involves an oversizing with respect to the energy required by the reference cycle but allows to exploit the supply or absorption of high currents for short intervals of time, provided that the fluctuations in the SOC are less than 5% (micro-cycles). An energy storage system of this type, subject to this type of stress, has an expected useful life of 10 ÷ 15 years [51,52,55]. Moreover, several studies have been carried out evaluating the use of degraded batteries, since they still have remaining capacity for grid support applications or emergency power supply [56–58]:

$$N\_{PAR\_B} = \max(N\_{PAR\_B}^P; N\_{PAR\_B}^E) \tag{27}$$

$$N\_{PAR\_{B}}^{E} = \frac{E\_B}{N\_{SE\_B} \cdot \mathcal{U}\_{CELL\_{B}} \cdot \mathcal{C}\_{AH\_{CEL\_{B}}} \cdot (\text{SOC}\_{\text{max}} - \text{SOC}\_{\text{min}})} \tag{28}$$

$$E\_B = \int\_{t\_0}^{t\_0 + T} P\_B(t)dt\tag{29}$$

$$N\_{PAR\_B}^P = \frac{P\_B^{\text{MAX}}}{N\_{SE\_B} \cdot P\_{CEIL\_B}^{\text{max}}} \tag{30}$$

where *Pmax CELLB* is the maximum power that the single electrochemical cell can deliver, determined in the discharge phase by using Equation (31), where *RDIS* [h−1] is the discharge C-rate, which is a measure of the rate at which the cell is discharged relative to its nominal capacity *CAHCELLB* [Ah]. The maximum power that the single cell can absorb in the charging phase is computed as a function of the charge C-rate *RCH* [h−1] [31]:

$$P\_{\text{CELL}\_{B\_{\text{DIS}}}} = R\_{\text{DIS}} \cdot C\_{AH\_{\text{CELL}\_{B}}} \cdot \mathcal{U}\_{\text{CELL}\_{B}} \tag{31}$$

$$P\_{\text{CELL}\_{B\_{CH}}} = R\_{\text{CH}} \cdot \mathbb{C}\_{Ah\_{\text{CELL}\_{B}}} \cdot \mathbb{U}\_{\text{CELL}\_{B}} \tag{32}$$

Given a commercial supercapacitor of rated voltage *UCELLSC* , the number of cells to be connected in series *NSESC* to obtain a given rated voltage of the supercapacitor pack *USC* is determined by:

$$N\_{SE\_{SC}} = \frac{U\_{SC}}{U\_{CELL\_{SC}}}\tag{33}$$

The number of supercapacitor branches to be connected in parallel, *NPARSC*, is determined through Equation (34), where *ESC*\_*max* is the maximum energy that the supercapacitor pack must deliver. It is highlighted that the voltage range is limited between VNOM/2 (SOCSC = 0) and VNOM (SOCSC = 1), delivering 75% of the stored energy:

$$N\_{PAR\_{SC}} = \frac{8}{3} \frac{E\_{SC\\_max}}{\left(N\_{SE\_{SC}} \cdot \mathcal{U}\_{CELL,SC}\right)^2} \cdot \frac{N\_{SE\_{SC}}}{C\_{CELL}} \tag{34}$$

$$E\_{\rm SC}(t\_p) = \int\_{t\_0}^{t\_0 + t\_p} [P\_{\rm SC}(t) - P\_{\rm SC}(t\_0)] dt \tag{35}$$

Finally, the total mass *WTOT* and volume *VTOT* linked to each energy storage system can be computed as follows:

$$
\mathcal{W}\_{TOT} = (1+\gamma) \cdot \mathcal{N}\_{SE} \cdot \mathcal{N}\_{PAR} \cdot \mathcal{W}\_{CELL} \tag{36}
$$

$$
\Delta V\_{TOT} = (1 + \delta) \cdot N\_{SE} \cdot N\_{PAR} \cdot V\_{CELL} \tag{37}
$$

where the coefficients γ and δ represent mass and volume rations of DC/DC converters and all the other additional elements necessary for the assembly and use of the energy storage system.
