## 2 DPF Model Considering Governor Delay of SG  
### Traditional DPF Model  
When the power system operates under the steady-state condition, the active power output of the SG (denoted as \(P_{\text{G}}\)) is balanced with the mechanical power supplied to the SG by the driving turbine (denoted as \(P_{\text{G,M}}\)). Upon the occurrence of the load disturbance, the unbalance between active power output and mechanical power of the SG yields the change of rotor speed, and simultaneously the change of frequency. To analyze the dynamic of the system frequency, the dynamic power flow (DPF) model is applied, which characterizes the dynamic change of the system frequency based on Equation (1).  
\[2\sum H_{\text{G}}f\frac{df}{dt}=P_{\text{acc}}=\sum P_{\text{G,M}}-\sum P_{ \text{L}}-P_{\text{loss}} \tag{1}\]  
where \(f\) is the system frequency, \(H_{\text{G}}\) is the inertia of the SG, \(P_{\text{G,M}}\) is the mechanical power supplied to the SG by the driving turbine, \(P_{\text{L}}\) is the active load, \(P_{\text{loss}}\) is the active power loss of the system, \(P_{\rm acc}\) is the accelerating power of the system, \(t\) is time, and \(d/dt\) is the differentiation to \(t\).  
As for the right-hand side of Equation (1), \(P_{\rm G,M}\) equals to \(P_{\rm G}\) under steady state operation, and in this case, \(\sum P_{\rm G,M}-\sum P_{\rm L}-P_{\rm loss}\) equals to \(\sum P_{\rm G}-\sum P_{\rm L}-P_{\rm loss}\), which is zero due to the electrical power balance of the system. When load disturbance occurs, the active power of SG (\(P_{\rm G}\)) instantly responds to the load variation, leading to unbalance between active power (\(P_{\rm G}\)) and mechanical power (\(P_{\rm G,M}\)) of the SG. This power unbalance is denoted as the accelerating power (\(P_{\rm acc}\)) in the dynamic power flow analysis [20], which is calculated with the right-hand side of Equation (1). The variation of the rotor speed is obtained based on the inertial response of the SG. Since the frequency of the SG is determined by the rotor speed, the dynamic frequency is evaluated based on Equation (1), which is transformed from the rotor motion equation of the SG.  
As to the description of the system frequency with the SGs in the dominant position, the average system frequency (ASF) model [18] is adopted, which defines the system frequency by the average value of the frequency of the SGs considering their respective inertias, i.e., for a system with \(n\) SGs, system frequency is defined as \(f=\sum\limits_{m=1}^{n}f_{m}H_{m}/\sum\limits_{m=1}^{n}H_{m}\).  
With the inertia release and the primary frequency control, the active power output of the SG is decided by the inertia, the static frequency characteristics, and the change of frequency, i.e., \(\Delta f\). For bus \(i\), the power constraints are given by,  
\[\left\{\begin{array}{l}\Delta P_{i}=\left(P_{\rm G,Mi0}-K_{\rm G,Mi}\Delta f \right)-\left(P_{\rm Li0}+K_{\rm Li}\Delta f\right)-\frac{H_{\rm Gi0}}{\sum H}P _{\rm acc}-P_{i}=0\\ \Delta Q_{i}=Q_{\rm Gi0}-Q_{\rm Li0}-Q_{i}=0\end{array}\right. \tag{2}\]  
where \(K_{\rm G,Mi}\) and \(K_{\rm Li}\) are the coefficients of the static frequency characteristics of the SG and the load, and the subscript \(0\) denotes pre-disturbance parameters.  
After calculating the system frequency based on Equation (1), the active power injected by the generators and the active power of the loads are modified considering the frequency change, and their modified values are then used for the power flow analysis. The obtained power flow results still adhere to the active power balance constraint. By solving (1) and (2) at each time step \(\Delta t\), the power flow distribution and system frequency will be found. Compared with the power flow model, the power constraint of the swing bus is added in the DPF model, and \(P_{\rm acc}\) is introduced to quantify the system frequency.