3.2.1. BESS Cost

A BESS is basically constituted by two components, the electrochemical battery and the system devoted to the power conversion. Thus, its installation cost can be split on a term proportional to the nominal power and another term relevant to the rated capacity. Regarding the share of BESS capacity reserved to DSO, several market mechanisms for remunerating this service are possible. In the paper, the simplest method has been supposed: the DSO refunds only the quote of the reserved share capacity. Regarding the cost term on power, proportional to the specific cost *cp* (in €/kW), it is entirely charged to the private investor because the nominal power is entirely free until the SoC does not overlap the reserved upper and lower semi-bands (Figure 2). Consequently, the initial investment (*IBESS*) for the private investor is defined as in (5):

$$\mathbf{I}\_{BESS} = \left(\mathbf{c}\_p \cdot \mathbf{P}\_n\right) + \left[\mathbf{c}\_t \cdot \left(\mathbf{C}\_n - \mathbf{C}\_{\rm DSO}\right)\right],\tag{6}$$

where *ce* is the unitary cost of capacity (expressed in €/kWh). *cp* and *ce* values depend on the BESS technology considered. The cost for the maintenance was assumed negligible in this paper even though it can be included in the model. If the BESS lifespan (*NBESS\_life*) is longer than the years' number of the planning period (*Nyears*), the residual value (*RBESS*) has to be considered and subtracted to the initial investment for a correct CBA:

$$\mathcal{L}\_{BESS} = I\_{BESS} - R\_{BESS} = \left(\mathbf{c}\_{p^\*} \mathbf{P}\_{\mathrm{n}}\right) + \left[\mathbf{c}\_{\mathrm{t^\*}} \left(\mathbf{C}\_{\mathrm{n}} - \mathbf{C}\_{\mathrm{DSO}}\right)\right] \cdot \left(1 - a^{N\_{\mathrm{gurn}}} \cdot \frac{\mathbf{N}\_{BESS\_{\mathrm{lfp}}} - \mathbf{N}\_{\mathrm{gurn}}}{\mathbf{N}\_{BESS\_{\mathrm{lfp}}}}\right),\tag{7}$$

where *a* = 1/(1 + *d*) is the actualization factor used to assess the net present value (NPV) of *RBESS*, derived by assuming a constant discount rate (*d*) during the whole planning period. On the contrary, if *NBESS\_life* < *Nyears*, the investment on a new storage device has to be included.

#### 3.2.2. Arbitrage

As aforementioned, arbitrage takes advantage of energy market price spreads (between off-peak and peak demand hours) that can produce value, even considering the efficiency of the BESS.

For each configuration of BESS proposed by the NSGA-II, the arbitrage term is calculated by subtracting the cost of storing energy in the device (charging phase) from the incomes of providing energy to the system (discharging phases), properly considering the round-trip efficiency of the BEES (i.e., the energy lost during the storage operation).
