## 2 LFC, INP, and CBRR
### Inp  
The basic principle of the INP operation and a steady-state correction value calculation with the INP for three CAs is given in [20]. A control-demand concept is used for the INP and a schematic block diagram representation is shown in Figure 3. Commonly, \(N\) CAs can be connected via virtual tie-lines, i.e., they can all activate the INP via the interchange factors \(K_{i}^{\rm INP}\), \(K_{i}^{\rm INP}\),..., \(K_{N}^{\rm INP}\) marked with the green rectangle. Note that the factor \(K_{j}^{\rm INP}\) represents the size of the INP interchange of the \(j\)-th CA in the \(i\)-th CA, where \(K_{j}^{\rm INP}=0\) determines that the possible INP interchange is equal to \(0\%\), whereas \(K_{j}^{\rm INP}=1\) determines that the possible INP interchange is equal to \(100\%\). Moreover, the values of \(K_{i}^{\rm INP}\), \(K_{j}^{\rm INP}\),..., \(K_{N}^{\rm INP}\) can be different. The cooperating CAs are connected to the \"red\" summator, forming virtual tie-lines, as shown in Figure 3. The input variables to the \"red\" summator are the demand powers of the cooperating CAs, i.e., \(P_{\rm di}\)  
Figure 3: Schematic block diagram representation of the LFC in the CA \(i\) (solid line) with the INP (solid and dotted line). Figure 2: Schematic block diagram representation of the LFC in the CA \(i\). \(P_{\rm dj}\),..., \(P_{\rm dN}\). The demand power of the CA \(i\) characterizes the maximum interchange power for the CA \(i\) among the cooperating CAs and is defined as  
\[P_{\rm di}=\Delta P_{\rm ei}-ACE_{i} \tag{3}\]  
according to [3; 29]. Introducing (1) and (2) in (3) gives the following relation:  
\[P_{\rm di}=\Delta P_{\rm Li}-B_{i}\Delta f_{i}. \tag{4}\]  
The power imbalance between generation and load in addition to \(K_{i}^{\rm INP}P_{\rm di}\) from the CA \(i\), \(K_{j}^{\rm INP}P_{\rm dj}\) from CA \(j\),..., \(K_{N}^{\rm INP}P_{\rm dN}\) from CA \(N\) is, for the CA \(i\), defined as  
\[ACE_{i}^{\rm INP}=B_{i}\Delta f_{i}+\Delta P_{i}+K_{i}^{\rm INP}P_{\rm di}-(K_{ j}^{\rm INP}P_{\rm dj}+,...,+K_{N}^{\rm INP}P_{\rm dN}). \tag{5}\]  
The output variables of the \"red\" summator are the correction powers of the cooperating CAs, i.e., \(P_{\rm cori}^{\rm INP}\), \(P_{\rm cori}^{\rm INP}\),..., \(P_{\rm corN}^{\rm INP}\), determined with a delay of \(T_{\rm s}\) due to the SH. The correction power of the CA \(i\) characterizes the maximum interchange power for the CA \(i\) among the cooperating CAs with a different sign of \(ACE_{i}\), and is included as  
\[ACE_{i}^{\rm INP}=(B_{i}\Delta f_{i}+\Delta P_{i})-P_{\rm cori}^{\rm INP}, \tag{6}\]  
where the terms in brackets denote \(ACE_{i}\). Moreover, only CAs with different signs of demand power, i.e., \(\rm sign(P_{\rm dj})\
eq\rm sign(P_{\rm dj})\), can net the demand for balancing energy. If two or more cooperating CAs are \"short\", then CBRR is used instead of the INP and vice versa [23]. Therefore, the cooperating CAs must be \"short\" and \"long\" in order to net the demand power through the INP. Hence, the balancing energy in CAs that net the balancing energy from the cooperating CAs can be reduced, and simultaneously the RR is released. The \(P_{\rm cori}^{\rm INP}\), \(P_{\rm cori}^{\rm INP}\),..., \(P_{\rm corN}^{\rm INP}\) is determined by the INP optimization module, considering numerous target functions, as given in [20]. Considering \(N\) CAs, then the \(P_{\rm cori}^{\rm INP}\) is, for the CA \(i\), expressed as  
\[P_{\rm cori}^{\rm INP}=-P_{\rm di}K_{i}^{\rm INP}+P_{\rm dj}K_{j}^{\rm INP}+, \ldots,+P_{\rm dN}K_{N}^{\rm INP}. \tag{7}\]  
Considering (4), then the \(P_{\rm cori}^{\rm INP}\) between \(N\) CAs is, for the CA \(i\), expressed as  
\[P_{\rm cori}^{\rm INP}=-(\Delta P_{\rm Li}-B_{i}\Delta f_{i})K_{i}^{\rm INP}+( \Delta P_{\rm Li}-B_{j}\Delta f_{j})K_{j}^{\rm INP}+,\ldots,+(\Delta P_{\rm LN }-B_{N}\Delta f_{N})K_{N}^{\rm INP}. \tag{8}\]  
In this way, the correction power of the CA \(i\) compensates the load variation that is varied by the frequency variation of the cooperating CAs.