## 2 LFC, INP, and CBRR
### Cgdr  
The basic principle of the CBRR operation and a steady-state correction-value calculation with the CBRR for three CAs is given in [22]. The same control-demand concept is used for the CBRR as is currently used for the INP, and a schematic block diagram representation is shown in Figure 4[3]. Similar to INP, \(N\) CAs can be connected via the virtual tie-lines, i.e., they can all activate the CBRR via the activation factors \(K_{i}^{\rm CBRR}\), \(K_{j}^{\rm CBRR}\),..., \(K_{N}^{\rm CBRR}\) marked with the green rectangle. Note that the factor \(K_{j}^{\rm CBRR}\) represents the size of the CBRR activation for the CA \(j\) in the CA \(i\), where \(K_{j}^{\rm CBRR}=0\) determines that the possible CBRR activation is equal to \(0\%\), whereas \(K_{j}^{\rm CBRR}=1\) determines that the possible CBRR activation is equal to \(100\%\). Moreover, the values of \(K_{i}^{\rm CBRR}\), \(K_{j}^{\rm CBRR}\),..., \(K_{N}^{\rm CBRR}\) can be different. The cooperating CAs are connected to the \"red\" summator, forming virtual tie-lines, as shown in Figure 4.  
The input variables to the \"red\" summator are the demand powers of the cooperating CAs, i.e., \(P_{\text{di}}\), \(P_{\text{dj}}\),..., \(P_{\text{dN}}\). The demand power of the CA \(i\) characterizes the maximum activation power for the CA \(i\) among the cooperating CAs, defined as (3), according to [3, 29]. Similar to INP, by introducing (1) and (2) in (3), (4) is obtained. The power imbalance between generation and load in addition to \(K_{i}^{\text{CBRR}}P_{\text{di}}\) from the CA \(i\), \(K_{j}^{\text{CBRR}}P_{\text{dj}}\) from the CA \(j\),..., \(K_{N}^{\text{CBRR}}P_{\text{dN}}\) from the CA \(N\) is, for the CA \(i\), defined as  
\[ACE_{i}^{\text{CBRR}}=B_{i}\Delta f_{i}+\Delta P_{i}-K_{i}^{\text{CBRR}}P_{ \text{di}}+(K_{j}^{\text{CBRR}}P_{\text{dj}}+,\ldots,+K_{N}^{\text{CBRR}}P_{ \text{dN}}). \tag{9}\]  
The output variables of the \"red\" summator are the correction powers of the cooperating CAs, i.e., \(P_{\text{cori}}^{\text{CBRR}}\), \(P_{\text{corj}}^{\text{CBRR}}\),..., \(P_{\text{corN}}^{\text{CBRR}}\), determined with a delay of \(T_{\text{s}}\) due to the SH. The correction power of the CA \(i\) characterizes the maximum activation power for the CA \(i\) among the cooperating CAs with matching sign of \(ACE_{i}\), and is included as  
\[ACE_{i}^{\text{CBRR}}=(B_{i}\Delta f_{i}+\Delta P_{i})+P_{\text{cori}}^{\text{ CBRR}}, \tag{10}\]  
where the terms in brackets denote \(ACE_{i}\). Moreover, only CAs with matching sign of demand power, i.e., \(\text{sign}(P_{\text{di}})\!=\!\text{sign}(P_{\text{dj}})\), can activate the demand for balancing energy. If any of the cooperating CAs are \"long\" and the others are \"short\", then INP is used instead of the CBRR and vice versa [21]. Therefore, the cooperating CAs must be either \"short\" or \"long\", depending on whether a positive or negative CBRR is activated. Hence, the balancing energy in CAs that activates the balancing energy in the cooperating CAs can be reduced, and simultaneously the RR is released. The \(P_{\text{cori}}^{\text{CBRR}}\), \(P_{\text{corj}}^{\text{CBRR}}\),..., \(P_{\text{corN}}^{\text{CBRR}}\) is determined by the CBRR optimization module, considering numerous target functions, as given in [22]. Considering \(N\) CAs, then the \(P_{\text{cori}}^{\text{CBRR}}\) is, for the CA \(i\), expressed as  
\[P_{\text{cori}}^{\text{CBRR}}\!=\!-P_{\text{di}}K_{i}^{\text{CBRR}}+P_{\text{ dj}}K_{j}^{\text{CBRR}}\!+,\ldots,+P_{\text{dN}}K_{N}^{\text{CBRR}}. \tag{11}\]  
Considering (4), then the \(P_{\text{cori}}^{\text{CBRR}}\) between \(N\) CAs is, for the CA \(i\), expressed as  
\[P_{\text{cori}}^{\text{CBRR}}\!=\!-(\Delta P_{\text{L}i}-B_{i}\Delta f_{i})K_{ i}^{\text{CBRR}}+(\Delta P_{\text{L}j}-B_{j}\Delta f_{j})K_{j}^{\text{CBRR}}\!+,\ldots,+(\Delta P_{\text{LN}}-B_{N}\Delta f_{N})K_{N}^{\text{CBRR}}. \tag{12}\]  
Similar to the INP, the correction power of the CA \(i\) compensates the load variation that is varied by the frequency variation of the cooperating CAs.