## 2 Battery Aging Following Frequency Regulation Cycles  
The first step of this work is to investigate the aging of one of the most promising battery technologies for supporting the frequency regulation of electrical networks, i.e., a lithium-manganese oxide battery. In order to achieve this objective, accelerating aging tests have been performed by the Italian transmission system operator Terna, by cycling the battery with very fast transitions from the discharge mode to the charge one by mimicking the stress suffered by the battery during the grid frequency regulation service. In fact, in real applications, the battery unit supplied power should be controlled on the basis of the instantaneous frequency value, i.e. discharging the battery when the network frequency falls below its threshold value and vice versa. Figure 1 shows a 24-h-long frequency variation in a real network bus [14]. The main electrical characteristics of the tested batteries are reported in Table 1.  
Figure 1: 24-h-long frequency variations in a real network bus. The aging test was carried out on a 3-kW battery module by performing 500 very stressful cycles, which lasted 10.2 h each. At the end of each cycle, the battery was completely recharged. In Figure 2 the voltage and current (the current profile follows a typical grid required power profile during the frequency regulation operation) measured during two aging cycles are reported and the recharge operations are highlighted. Once all the aging cycles were performed, the battery capacity was reduced by 7.5%. Figure 3 shows the battery capacity decrease during the aging test on the basis of the cycles. The anomalous capacity rise during the test, which is visible in Figure 3, is chiefly due to the fact that the battery State of Charge (SoC) is calculated by means of the battery management system (BMS), starting from the integral of the charge/discharge current and by subsequently comparing the current integral with the battery capacity. Each BMS needs to be periodically re-calibrated to correct the current integration \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{1}{c}{**Rank Characteristics**} & \multicolumn{2}{c}{**Module Characteristics**} \\ \hline Number of modules & 16 module + 1 BMS & Number of cells & 16 + 1 BMS \\ Dimensions & 1049 \(\times\) 549 \(\times\) 1851 mm & Dimensions & 485 \(\times\) 510 \(\times\) 162.5 mm \\ Weight & 1000 kg & Weight & 48 kg \\ Capacity & 60 Ah & Capacity & 60 Ah \\ Rated voltage & 947.2 V & Rated voltage & 59.2 V \\ Nominal Energy & 47.7 KWh & Nominal Energy & 2.98 kWh \\ Operating voltage range & 768–1054.7 V & Operating voltage range & 48–65.92 V \\ Rated discharge time & 1 h & Rated discharge time & 1 h \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Electrical characteristics of the tested technology. Figure 2: Measured voltage (**a**) and current (**b**) during two aging cycles. errors but during the tests no BMS re-calibration was performed, except in case of anomalous behaviour of the batteries. In order to identify a proper strategy to limit this aging effect, an electrical model of a battery module was developed in the Matlab Simulink environment, by using the Thevenin second order dynamic modelling approach described in [21], whose circuital representation is shown in Figure 4. The model parameters are inferred from specific voltage experimental measurements. In particular, the computation of the equivalent parameters is performed by exploiting the battery relaxation period after a discharge (see Figure 4, where the discharge phase starts at 125 s) at different battery SoCs, temperatures \(\theta\) and currents \(i\). In fact, with reference to Figure 4, the battery voltage variation during the time interval from \(t_{0}\) to \(t_{2}\) can be represented by means of two \(R\)-\(C\) circuits in series. The voltage behaviour in \(t_{0}\) can be modelled by means of a resistance \(R_{0}\).