## 2 Sharing State-of-Charge (SOC) Information among Grid-Forming Units without Communication  
To respond quickly to load changes, an ESS is proposed to be used as a grid-forming unit. Many studies have used ESSs as grid-feeding units for primary control [8; 9; 10; 11], but a grid-forming unit reacts much faster than a grid-feeding unit and is able to share transient load changes, since a grid-forming unit is controlled as a voltage source [35]. By referring to [35], grid-forming control acts as a fixed-voltage source, and grid-feeding acts as a fixed-current source. For sharing a load change, conventional _P-f_ droop control is used, and the relationship between the output active power and frequency of the grid-forming unit can be expressed as:  
\[f_{o}=f_{nom}-m\Big{(}P_{o}-P_{ref}\Big{)} \tag{1}\]where \(f_{nom}\) and \(P_{ref}\) correspond to the nominal and reference values for frequency and output active power, respectively, and \(f_{o}\) and \(P_{o}\) correspond to the measured output frequency and active power, respectively. The coefficient \(m\) denotes the droop coefficient and is usually determined as [36]:  
\[m=\frac{f_{nom}-f_{min}}{P_{max}-P_{min}} \tag{2}\]  
where \(f_{min}\) is the allowable minimum value of the system frequency, and \(P_{max}\) and \(P_{min}\) are the maximum and minimum output power of a DG, respectively. \(f_{min}\) should be the same for all grid-forming units and \(P_{max}\) and \(P_{min}\) can be distinguished based on their power ratings and generation costs, etc. By adjusting the droop coefficients among grid-forming units, the proportion of load sharing can be determined as desired at the primary control level. The main reason that ESSs have not been used as grid-forming units is that they cannot manage their own SOC, since a grid-forming type unit cannot control its own output power. To solve this problem, the output frequencies of all grid-forming units are adjusted to reflect the rough information of the SOC--i.e., whether the ESS should be charged or discharged--with respect to the frequency. To change the output frequencies of the grid-forming DGs almost simultaneously, other types of grid-forming DGs should know the SOC level of the grid-forming ESS. To achieve this without any communication devices at the secondary control level, it is assumed that all the hourly output power set points are predetermined, and the output power set point and droop coefficient of the ESS are given for every grid-forming DG. With those values known, other grid-forming DGs can estimate the output power of an ESS according to the following equations:  
\[f_{o,ESS}=f_{nom}-m_{ESS}\big{(}P_{o,ESS}-P_{ref,ESS}\big{)} \tag{3}\]  
\[f_{o,DG}=f_{nom}-m_{DG}\big{(}P_{o,DG}-P_{ref,DG}\big{)} \tag{4}\]  
where the subscripts ESS and DG denote grid-forming ESS and another type of grid-forming DG, respectively. From the perspective of the ESS, \(m_{DG}\) and \(P_{ref,DG}\) are given. Since \(f_{o,DG}\) is identical to \(f_{o,ESS}\), Equations (3) and (4) can be combined as:  
\[m_{ESS}\big{(}P_{o,ESS}-P_{ref,ESS}\big{)}=m_{DG}\big{(}P_{o,DG}-P_{ref,DG} \big{)} \tag{5}\]  
Equation (5) can be rearranged in terms of \(P_{o,ESS}\) as:  
\[P_{o,ESS}=\frac{m_{DG}}{m_{ESS}}\big{(}P_{o,DG}-P_{ref,DC}\big{)}+P_{ref,ESS} \tag{6}\]  
From the perspective of grid-forming DG, all variables on the right-hand side of Equation (6) are known, so the output power of the grid-forming ESS can be calculated by Equation (6). The SOC of the grid-forming ESS can be expressed as [37]:  
\[SOC_{ESS}=SOC_{ESS,init}-\int\frac{P_{o,ESS}}{CAP_{ESS}}dt \tag{7}\]  
where \(SOC_{ESS}\) is the SOC of grid-forming ESS, \(SOC_{ESS,init}\) is the initial value of \(SOC_{ESS}\), and \(CAP_{ESS}\) is the capacity of the grid-forming ESS. Assuming that \(SOC_{ESS,init}\) and \(CAP_{ESS}\) are known for every grid-forming DG, grid-forming DGs can estimate the SOC of the grid-forming ESS by using local parameters Equations (6) and (7), without any communication infrastructure.