## 2 LFC, INP, and CBRR
### Simultaneous Operation of the INP and the CBRR  
In the cooperating CAs, \(\Delta P_{\text{Li}}\) changes randomly and continuously. Consequently, cases of \(\Delta P_{\text{Li}}\) sign changes can occur, resulting in a continuous switching between the INP and the CBRR, which causes undesirable \(\Delta P_{\text{ori}}\) sign changes. Such a situation might occur when the signs of \(\Delta P_{\text{di}}\) and \(\Delta P_{\text{dj}}\) are changed with a short time delay. Therefore, a function for \(P_{\text{ori}}\) adjustment is proposed as one of the contributions of this article.  
A schematic block diagram representation for \(P_{\text{ori}}\) adjustment in relation to \(P_{\text{di}}\) of the CA \(i\) is shown in Figure 5. The signs of the two successive samples, i.e., \(P_{\text{di},k-1}\) and \(P_{\text{di},k}\), are compared with the relational operator, whose output is connected to a switch, marked with \"c\". Two states are possible, i.e.,  
* State 1: \(\text{sign}(P_{\text{di},k-1})\!=\!\text{sign}(P_{\text{di},k})\) and
* State 2: \(\text{sign}(P_{\text{di},k-1})\!\
eq\!\text{sign}(P_{\text{di},k})\).  
For State 1, the switch position is \"1\" and the output variable is \(P_{\text{cor}i}^{\Delta\text{Dj}}=P_{\text{cor}i}\). For State 2, the switch position is \"2\" and the output variable is \(P_{\text{cor}i}^{\Delta\text{Dj}}=0\).  
The implementation of the \(P_{\text{cor}i}\) adjustment in the INP and the CBRR framework is shown in Figure 6. The simultaneous operation of the INP and the CBRR, considering the \(P_{\text{cor}i}\) adjustment, can be described using the pseudo-code, as shown in Algorithm 1.  
An example of the simultaneous operation of the INP and the CBRR, with and without the \(P_{\text{cor}i}\) adjustment, is shown in Figure 7. The values of the loads were set in such a way that the simultaneous operation of the INP and the CBRR was possible. In the time interval of 80-100 s, the INP operated between CA\({}_{1}\)-CA\({}_{3}\) and CA\({}_{2}\)-CA\({}_{3}\), whereas the CBRR operated between CA\({}_{1}\)-CA\({}_{2}\). During this time interval, \(P_{\text{cor}i}\) due to the INP was possible in all three CAs, whereas \(P_{\text{cor}i}\) due to CBRR was possible only in CA\({}_{1}\) and CA\({}_{2}\). At \(t\!=\!100\) s, a simultaneous step change of the load in CA\({}_{1}\) and CA\({}_{3}\) was applied, whereas, in CA\({}_{2}\), it was applied with a delay, i.e., at \(t\!=\!100.05\) s, as seen at \(t\!=\!102\) s due to SH with \(T_{\text{s}}\!=\!2\) s. Time responses of \(P_{\text{di}}\) and \(P_{\text{cor}i}\) at \(t\!=\!100\) s with one step-size activation of the INP in CA\({}_{1}\) and CA\({}_{3}\), and the CBRR in CA\({}_{3}\) is clearly seen in Figure 7a (without \(P_{\text{cor}i}\) adjustment). Note that \(P_{\text{cor}i}\) due to CBRR should be zero in CA\({}_{3}\). Additionally, at \(t\!=\!102\) s, one step-size  
Figure 5: Schematic block diagram representation for \(P_{\text{cor}i}\) adjustment in the CA \(i\).  
Figure 6: Schematic block diagram representation for \(P_{\text{cor}i}\) adjustment with the INP and the CBRR in the CA \(i\).  
activation of the CBRR in CA\({}_{2}\) is also seen. However, in Figure 7b (with \(P_{\text{cori}}\) adjustment), at \(t=\)100 s and \(t=\)102 s, the value of \(P_{\text{cori}}\) was zero in all CAs. In this way, delayed \(P_{\text{di}}\) sign changes have no impact on the switching between the INP and the CBRR. Moreover, the \(P_{\text{cori}}\) variation is significantly reduced using the proposed adjustment.