## 2 Materials and Methods  
Given the pivotal need for new resources for the grid's frequency control, as detailed in the previous chapter, this study focuses on the use of a BESS numerical model to analyze FCR provision. In particular, the goal is to investigate the best option for the management of the BESS SoC. For such a goal, a reliable BESS model is required in order to provide a realistic figure; in the proposed study a runtime SoC evolution empirical model developed on experimental data by a large-scale industrial BESS [23] is adopted. It features a variable efficiency depending on the power rate and SoC. The BESS model receives as input power setpoints requested grid side per step of the simulation. The power setpoints are updated via a lookup table that estimates the step efficiency of the system. This lookup table considers the instantaneous SoC and power requested to the battery and returns the efficiency of the system, considering PCS and battery losses. The efficiency is multiplied (charging) or divided (discharging) by the power setpoint to assess the power flowing in the battery. This power is checked against the maximum/minimum power that can be provided by a battery at that SoC. To do this, the model features a capability curve that prevents the battery from delivering high power rates at extremely high (for charging) or low (for discharging) SoC. If power requested is outside the capability of the battery, the absolute value of real power is set equal to the limit. The real power is the responsibility for the SoC variation in the step. The model computes SoC evolution each step to provide the power requested. Moreover, the model returns if the dispatching order is not (or partially) respected. This model is implemented in a MATLAB Simulink(tm) numerical tool. The model is fed by network frequency data from Central Europe Synchronous Area (CESA) for the year 2016 [27]. These data have a 1 s sampling rate and a resolution of 0.1 MHz. The FCR controller block of the Simulink(tm) tool transforms frequency trends in power setpoints requested to BESS via a droop control curve based on the following equation. \[\mathrm{D}=-(\mathrm{dF/F_{n}})/(\mathrm{dP/P_{n}})\times 100, \tag{1}\]  
where D is the droop value in percentage, dF is the frequency deviation in Hz, F\({}_{n}\) is the nominal frequency of 50 Hz, dP is the power setpoint requested for FCR in MW, P\({}_{n}\) is the nominal power of the BESS in MW. In this study, the convention of generators is adopted, with power injected to the grid as positive power and power absorbed as negative. In addition, the curve features a dF dead-band inside which dP requested is equal to 0 and a full activation dF over which the dP requested is equal to P\({}_{n}\). This droop curve is an elaboration of the one in place in Italy for conventional thermal generators [9; 28]. In the framework of the study, D = 0.075%. All the relevant data for the curve can be seen in Table 1. The curve is shown in Figure 1. \begin{table}
\begin{tabular}{c c} \hline
**Drop Value (D)** & **0.075\%** \\ \hline Dead-band dF (mHz) & \(\pm\)20 \\ Full activation dF (mHz) & \(\pm\)37.5 \\ \hline \end{tabular}
\end{table} 
Table 1: Droop curve data. Besides the FCR controller, another block implements the SoC restoration strategy in the Simulink(tm) tool. The power requested by FCR provision (\(\mathrm{P_{FCR}}\)) and the one required for SoC restoration (\(\mathrm{P_{rest}}\)) are summed up and represent the power setpoint demanded to the BESS (\(\mathrm{P_{req}}\)). \[\mathrm{P_{req}}=\mathrm{P_{FCR}}+\mathrm{P_{rest}}. \tag{2}\]  
The BESS model updates the power setpoint consistently with efficiency and capability charts, and consequently updates SoC value for each second of the analyzed year. The outcomes of the model are the SoC trend, the power and energy flows requested and actually flowing in the battery, the share of energy flows for FCR provision and for SoC restoration.