## 2 Frequency and Active Power Control Schemes
### Proposed Method
SOC control is only activated when the SOC range limits are approached, as shown in Figure 2. Minimum (\(SOC_{bi,min}-SOC_{bi,min+}\)) and maximum (\(SOC_{bi,max-}-SOC_{bi,max+}\)) zones are defined, allowing gradual activation of the SOC regulation module, according to coefficients \(c_{bi,up}\) (green) and \(c_{bi,dn}\) (red). As illustrated in Figure 2, \(c_{bi,up}>0\) while \(SOC_{bi}<SOC_{bi,min+}\) and \(c_{bi,dn}>0\) while \(SOC_{bi}>SOC_{bi,max-}\). The term \(P^{*}_{bi,soc}\) is then superimposed on the output of the frequency regulation module \(P^{*}_{bi,f}\) to prevent BESS under/over-charging. This imposed deviation of BESS output from the AGC command, as well as \(c_{bi,(up/dn)}\), are fed back into the AGC to modify the participation factors of all units according to Equations (1)-(3), thus preventing any impact on secondary frequency regulation. Equations (1)-(3) are used distinctly for upwards participation factors' calculations (using the subscript \"\(up\)\"), and downwards participation factors' calculations (using the subscript \"\(dn\)\"). The upwards (downwards) BESS aFRR participation factor is affected only by \(c_{bi,up}\) (\(c_{bi,dn}\)), since high (low) BESS energy reserves do not restrict its upwards (downwards) secondary reserve provision capability. \[p\!f_{bi,(up/dn)}=\frac{(1-c_{bi,(up/dn)})p\!f_{bi,ED,(up/dn)}}{1-c_{bi,(up/dn )}p\!f_{bi,ED,(up/dn)}},\,\forall bi\in ESS \tag{1}\]\[pf_{gi,(up/dn)}=(1-\sum_{bi\in ESS}pf_{bi,(up/dn)})\frac{pf_{gi,ED,(up/dn)}}{ \sum\limits_{gi\in G}pf_{gi,ED,(up/dn)}},\ \forall gi\in G \tag{2}\]  
\[pf_{gi,noess,(up/dn)}=\frac{pf_{gi,ED,(up/dn)}}{\sum\limits_{gi\in G}pf_{gi,ED,(up/dn )}},\ \forall gi\in G \tag{3}\]  
In the simulations presented in Section 5, it is assumed that the state of charge of each storage unit should be maintained within 20-90% (similarly as in [14; 39; 52; 56]), thus \(SOC_{bi,min+}\) and \(SOC_{bi,max+}\) are set at these values (see Table 10). Selection of \(SOC_{bi,min+}\) and \(SOC_{bi,max-}\) parameters defines the SOC edge intervals where the battery has only partial contribution to frequency regulation. Selection of relatively high \(SOC_{bi,min+}\) and low \(SOC_{bi,max-}\) values will reduce battery degradation and provide a smoother response in SOC edge intervals, but at the same time will result in under-exploitation of BESS regulation capabilities and reserves. In this work, a 5% range is considered sufficient, thus \(SOC_{bi,min+}\) and \(SOC_{bi,max-}\) are set to 25% and 85%, respectively. A parametric evaluation is presented in Section 5.2.2, to show the effect of \(SOC_{bi,min+}\) on the performance of the proposed control method. Given the slow dynamics of SOC change, a simple and straightforward tuning process is proposed for \(K_{bi}\): With reference to Figure 1, a SOC level is defined where \(P^{*}_{bi,soc}\) will always cancel \(P^{*}_{bi,f}\), to exclude discharging below that SOC. To observe this requirement, \(K_{bi}\) is selected such that \(P^{*}_{bi,soc}=-1\) pu at the specified SOC level. For example, if \(SOC_{bi,min-}=20\%\), \(SOC_{bi,min+}=25\%\), and half of the SOC control activation zone (2.5%) is used as a safety margin against hitting \(SOC_{bi,min-}\), then \(P^{*}_{bi,soc}=-1\) pu when SOC drops to 22.5%. Assuming \(SOC^{*}_{bi}=55\%\), as noted in Section 3, this (i.e., \(P^{*}_{bi,soc}=-1\) pu for \(SOC_{bi}=22.5\%\)) is achieved by setting \(K_{bi}=\frac{1}{c_{bi,up}(SOC^{*}_{bi}-SOC_{bi})}=\frac{1}{0.5(55-22.5)}=0.0615\), which is the value used in the simulations in Section 5. The same tuning process can be followed around maximum SOC. A \(K_{bi}\) parametric analysis is presented in Section 5.2.1. The implementation of the proposed method requires that at least one more regulating unit is on-line to undertake part of the aFRR provided from the BESS, according to Equations (2) and (3), whenever its energy variation margins are depleted (\(SOC_{bi}\) < \(SOC_{bi,min+}\) or \(SOC_{bi}\) > \(SOC_{bi,max-}\)).