## 2 Materials and Methods
### Techno-Economic Analysis  
The results of the model are analyzed in terms of energy flows and associated revenue streams. The energy requested for FCR (\(\mathrm{E_{FCR}}\)) is computed as the summation of the absolute value of \(\mathrm{P_{FCR}}\) in time. The energy for SoC restoration is computed from \(\mathrm{P_{rest}}\) both for charge (\(\mathrm{E_{ch}}\)) and discharge (\(\mathrm{E_{dis}}\)). The energy non provided (\(\mathrm{E_{NP}}\)), relevant for the estimation of NPP, is computed by the model. NP occurs when the battery is not able to provide with accuracy the energy requested for the regulation. In case the error between \(\mathrm{P_{FCR}}\) and the power actually delivered (\(\mathrm{P_{del}}\)) by BESS overcomes 5%, then all the power requested by FCR for that instant is considered as nonprovided (\(\mathrm{P_{NP}}\)). \[\mathrm{P_{NP}} =\mathrm{P_{FCR}}\quad\text{if }\big{|}\mathrm{P_{req}}-\mathrm{P_{del}} \big{|}/\mathrm{P_{req}}>5\% \tag{9}\] \[\mathrm{P_{NP}} =0\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\]  
where \(\mathrm{E_{NP}}\) is the summation of the absolute values of \(\mathrm{P_{NP}}\). NP is eventually \(\mathrm{P_{NP}}\) expressed as a percentage of \(\mathrm{E_{FCR}}\). This index measures the reliability of the BESS as a BSP with different schemes. The cash flows related to FCR provision, to SoC restoration and to NPP are computed valorizing the energy flows with the indexes reported in Table 4, mainly referred to the Italian market, except for FCR remuneration which is referred to the German market (in \(\mathrm{\upxi}\)/MW/week). However, according to [34], also in Italy, FCR could be remunerated by means of availability payments in \(\mathrm{\upxi}\)/MW/week in the near future. The energy necessary to charge BESS for SoC restoration is valued at the yearly average purchase price in the Italian DAM, whereas the discharging energy at the yearly average positive imbalance price. NPP is valued at the average yearly marginal price in the Italian BM (to date, NPP for FCR is not defined in Italy, since FCR is mandatory). The selected parameters aim to be representative of the current situation and possible evolution in Continental Europe. Figure 8: The double threshold domain for SoC restoration. The red area identifies the condition of (dF, SoC) for triggering the flag variable and starting SoC restoration. The pink area is where the offsetting can be exploited for charging towards SoCtarget. The green area shows where \(\mathrm{P_{req}}\) can offset to discharge. The SoC restoration strategies allowed by the DoFs influence the return of the investment on BESS, too. To give a figure of this, the internal rate of return (IRR) for the investment at 5 y is computed as follows. \[\mathrm{NPV}=\mathrm{CAPEX+}\sum\
olimits_{i}^{N}\frac{\mathrm{NCF}}{\left(1+ \mathrm{IRR}\right)^{i}}+\frac{\mathrm{RV}}{\left(1+\mathrm{IRR}\right)^{N}}=0, \tag{10}\]  
\[\mathrm{CAPEX}=\mathrm{k_{e}}\times\mathrm{E_{n}}+\mathrm{k_{p}}\times( \mathrm{P_{n}}-\mathrm{E_{n}}), \tag{11}\]  
\[\mathrm{NCF}=\mathrm{R_{FCR}}+\mathrm{C_{ch}}+\mathrm{R_{dis}}+\mathrm{NPP}, \tag{12}\]  
where CAPEX is the cost investment for the BESS, based on the size of BESS and parameterized over \(\mathrm{k_{e}}=400\) k\(\epsilon\)/MWh and \(\mathrm{k_{p}}=150\) k\(\epsilon\)/MW [38]; N is the time horizon for the investment, set at 5 y; NCF is the net cash flow as the sum of the previously described revenue and cost streams (see Equation (12)); RV is the residual value of the asset at the end of year N computed as in Equation (13). \[\mathrm{RV}=\mathrm{CAPEX}\times(\mathrm{t_{EoL}}-(\mathrm{N}+1))/\mathrm{t_{ EoL}}, \tag{13}\]  
where \(\mathrm{t_{EoL}}\) is the expected end of life (EoL) in years, hypothesizing linear RV decay with time. To estimate battery life, a SoH model was developed based on [26]. The SoH model considers both cycle aging and calendar aging in terms of capacity fade.