# Research on Robust Day-Ahead Dispatch Considering Primary Frequency Response of Wind Turbine
## 3 Optimization Model Considering Primary Frequency Response
### Primary Frequency Response  
We consider that in the real-time market, the wind turbines using VSG technology could provide an inertial response and PFR. The method employed to determine the reserve capacity through the fixed reserve capacity or a certain proportion of the wind power output is not a widely applicable one. Therefore, we will establish a more detailed primary reserve model with the optimization dispatch framework, in accordance with the VSG technical standards and system frequency requirements. According to the grid operation code, there is a certain lower limit of the system frequency (this paper only considers the frequency drop). Taking the state grid as an example, the rated frequency is 50 Hz, with a frequency offset of 0.2-0.5 Hz. For the 5-30 s PFR time frame, the power adjustment after the frequency drop involves a power loss, system inertia, ramp response rate of the governor and frequency dead band. Based on this, we depict the governor model as in Figure 1:where \(P_{m}(t)\) and \(P_{e}(t)\) represent the mechanic and electromagnetic power of the system; \(P_{loss}\) is the loss of the active output caused by a fault in the units; \(R_{i}^{F}\) is the primary reserve of the unit \(i\), while \(C_{N}\) is the ramp rate of the governor; \(f_{0},f_{d},f_{N}\) and \(f_{M}\) represent the initial frequency, the governor's dead band, the nadir-based frequency of the unit after the fault, and the lower limit of the system frequency, respectively; and \(t_{d}\) and \(t_{N}\) are the time when the frequency reaches the dead band and the lowest point. The active output decrease can result in a decreased system frequency. At the same time, the governor responds within the range of the reserve. The mechanical power of the system increases, which leads to the balance of the mechanical power and the electromagnetic power. If we ignore the load damping, the relationship between the frequency variation and power could be shown as follows [32; 33]:  
\[M_{H}\frac{df(t)}{dt}=-\Delta P=P_{m}(t)-P_{e}(t) \tag{4}\]  
where \(M_{H}\) indicates the system inertia after the fault with the MWs/Hz unit; and through the integration of both sides of the Equation (4), we get:  
\[\int\limits_{0}^{t_{N}}\frac{df(t)}{dt}dt= f_{N}-f_{0}=\frac{1}{M_{H}}\int\limits_{0}^{t_{N}}[P_{m}(t)-P_{e}(t) ]dt=-\frac{1}{M_{H}}\bigg{(}P_{loss}t_{d}+\frac{P_{loss}^{2}}{2C_{N}}\bigg{)} \tag{5}\]  
Using the governor model in Figure 1, Equation (5) equals:  
\[f_{N}-f_{0}=-\frac{1}{M_{H}}\bigg{(}P_{loss}t_{d}+\frac{P_{loss}^{2}}{2C_{N}} \bigg{)} \tag{6}\]  
If \(C_{M}\), the maximum ramp rate of the governor, exists, \(f_{M}\) corresponds to it. \(f_{N}\) and \(C_{N}\) could be replaced by \(C_{M}\) and \(f_{M}\); then, Equation (6) would become:  
\[C_{M}=\frac{0.5P_{loss}^{2}}{M_{H}(f_{0}-f_{M})-P_{loss}t_{d}}\frac{M_{H}= \frac{P_{loss}t_{d}}{f_{d}}}{M_{H}(f_{0}-f_{M}-f_{d})} \tag{7}\]  
The power fluctuation of the system is supported by inertia for a short time within the dead band of the governor, so the system inertia time constant can be expressed by Equation (7): \(M_{H}=\frac{P_{loss}t_{d}}{f_{d}}\). Figure 1: Governor Model. Therefore, if we only consider the synchronous generator in the system, for \(P_{\text{loss}}\), the primary reserve of the system satisfies the following restraints:  
\[\sum R_{i}^{F}\geq P_{\text{loss}} \tag{8}\]  
\[0\leq R_{i}^{F}\leq 2C_{i}\frac{M_{H}(f_{0}-f_{M}-f_{d})}{P_{\text{loss}}} \tag{9}\]  
where \(C_{i}\) is the ramp rate of the governor of unit \(i\). From Equations (8) and (9), we can see that when the power imbalance is small, a sufficient reserve capacity can be retained in the optimization dispatch through the above constraints. Equation (8) indicates that the primary reserve of the system is sufficient to cover the power loss and maintain the power balance, while through Equation (9) we determine the reserve capacity based on the ramp rates of different governors.