## 2 Emergency State and Automatic Generation Control
### Control Objective of Automatic Generation Control  
In each control area, the AGC controller aims to (i) minimize the systemic frequency deviation \(\Delta f\), (ii) reduce the value of the ACE \(e_{\rm ACE}\), and (iii) maximize the CPS index \(k_{\rm CPS}\).  
The value of the ACE \(e_{\rm ACE}\) can be calculated as  
\[e_{\rm ACE}=\Delta P_{\rm t}-10B\Delta f, \tag{1}\]  
where \(\Delta P_{\rm t}\) is the scheduled tie-line power deviation; \(B\) is the frequency response coefficient of the control area (in MW/0.1 Hz); \(\Delta f\) is the frequency deviation (in Hz).  
The CPS index \(k_{\rm CPS}\), which includes CPS1 index and CPS2 index, is established by the _North American Electric Reliability Council_ (NERC) [12; 34; 35]. The CPS index is the statistic index of \(\Delta f\) and \(e_{\rm ACE}\) over a long period of time, rather than the real-time values of \(\Delta f\) and \(e_{\rm ACE}\). The CPS index \(k_{\rm CPS}\) can be calculated as  
\[k_{\rm CPS}(\%)=\left(\frac{N_{a_{\rm CF}=1}}{N_{a_{\rm CF}=1}+N_{a_{\rm CF} =0}}\right)\times 100\%, \tag{2}\]  
Figure 1: The three gradations of frequency control.  
where \(N_{a_{\text{CF}}=1}\) is the number of periods when \(\alpha_{\text{CF}}=1\); \(N_{a_{\text{CF}}=0}\) is the number of periods when \(\alpha_{\text{CF}}=0\). The variable \(\alpha_{\text{CF}}\) can be calculated as  
\[\alpha_{\text{CF}}=\left\{\begin{array}{l}1,\left\{\begin{array}{l}\alpha_{ \text{CF1}}\geq 200\%\\ 100\%\leq\alpha_{\text{CF1}}\leq 200\%,\alpha_{\text{CF2}}=1\\ 0\end{array}\right.,\\ \alpha_{\text{CF1}}<100\%\\ 100\%\leq\alpha_{\text{CF1}}\leq 200\%,\alpha_{\text{CF2}}=0\end{array} \right.\end{array}\right., \tag{3}\]  
where \(\alpha_{\text{CF1}}\) and \(\alpha_{\text{CF2}}\) can be calculated as follows.  
\[\alpha_{\text{CF1}}=\frac{E_{\text{AVE-1 min}}\Delta F_{\text{AVE-1 min}}}{-10B \varepsilon_{\text{1 min}}^{2}}, \tag{4}\]  
\[\alpha_{\text{CF2}}=\frac{E_{\text{AVE-10 min}}}{16.5e_{\text{10 min}}\sqrt{B_{\text{S}}B}}, \tag{5}\]  
where \(E_{\text{AVE-1 min}}\) is the clock-1-min average of the ACE; \(E_{\text{AVE-10 min}}\) is the the clock-10-min average of the ACE; \(\Delta F_{\text{AVE-1 min}}\) is the clock-1-min average of the frequency deviation; \(\varepsilon_{\text{1 min}}\) is the targeted frequency bound for the CPS1 index with clock-1-min; \(\varepsilon_{\text{10 min}}\) is the targeted frequency bound for the CPS1 index with clock-10-min; \(B\) represents the frequency bias of the control area, expressed in MW/0.1 Hz; \(B_{\text{S}}\) represents the frequency bias of the power system, expressed in MW/0.1 Hz.  
Furthermore, the CPS1 index evaluates the impact of the ACE deviation on the frequency of system, while the CPS2 index is used to restrict the ACE magnitude. They can be calculated as follows:  
\[k_{\text{CPS1}}=(2-\left\{\alpha_{\text{CF1}}\right\}_{T})\times 100\%, \tag{6}\]  
\[k_{\text{CPS2}}=(\frac{N_{a_{\text{CF2}}<1}}{N_{a_{\text{CF2}}<1}+N_{a_{\text {CF2}}\geq 1}})\times 100\%, \tag{7}\]  
where \(\left\{\alpha_{\text{CF1}}\right\}_{T}\) is the average value of the variable \(\alpha_{\text{CF1}}\) at the period of \(T\), which is always 1 year; \(N_{a_{\text{CF2}}<1}\) is the number of periods when \(\alpha_{\text{CF2}}<1\); \(N_{a_{\text{CF2}}\geq 1}\) is the number of periods when \(\alpha_{\text{CF2}}\geq 1\).