## 4 A Unified Frequency Model for Multi-Machine Systems  
For a hydro-PV complementary system with multiple hydropower generators and PV generators, it is necessary to construct a unified model that can consider all generators. Due to the uniform frequency of the whole grid during ULFO, the frequency deviations of all generators remain consistent, that is, the rotor motion equation of the \(i\)-th generator is shown in the first equation of Equation (1). Assuming that there are \(n\) hydropower generators in the system, since all the speed deviations are the same, the equations of motion of all units can be linearized and then added together to obtain:  
\[\sum_{i=1}^{n}T_{ji}\frac{d\Delta\omega}{dt}=\sum_{i=1}^{n}\Delta P_{mi}-\sum_ {i=1}^{n}\Delta P_{ei}-\sum_{i=1}^{n}D_{i}\Delta\omega \tag{20}\]  
Since the mechanical power deviation (\(\Delta P_{m}\)) of a hydropower generator is obtained by the speed deviation (\(\Delta\omega\)) through the two control links of the governor and the turbine, the above equation can be converted into a transfer function block diagram to obtain a unified frequency model, as shown in Figure 8.  
After strictly adding all the motion equations, according to the research needs of ULFO, an approximate condition is introduced:  
1. Neglecting network loss, it is considered that the electromagnetic power of all generators is approximately equal to the load power, that is \(\Sigma\Delta P_{e}=\Sigma\Delta P_{L}\);
2. Neglecting the frequency and voltage regulation effects of the load, it is considered to be a constant impedance load, that is \(\Sigma\Delta P_{e}=\Sigma\Delta P_{L}=0\).  
When the active power additional controller of the PV generator is introduced, the PV generator participates in the frequency adjustment process through the controller, as shown in Figure 7. It is necessary to consider the impact of PV active output deviation on system electromagnetic power deviation when analyzing the influence of the control task, that is \(\Sigma\Delta P_{e}=\Sigma\Delta P_{L}-\Sigma\Delta P_{PV}\). According to the approximate condition, load deviation is considered to be 0, that is \(\Sigma\Delta P_{e}=-\Sigma\Delta P_{PV}\), so it is necessary to add the additional controller and power controller of the PV generator to the unified frequency model.  
Figure 8: A unified frequency model for multiple hydropower generators.  
After considering both hydro and PV controllers, the unified frequency response model of the multi-machine system considering \(n\) hydropower generators and \(m\) PV generators is shown in Figure 9.  
It is worth noting that the unified frequency model constructed in the above figure is only suitable for the analysis of ULFO, because only ULFO is characterized by the uniform oscillation frequency of the whole network. Since the oscillation mode has nothing to do with the excitation system of the hydroelectric unit, it can only be used to consider the prime mover system and the hydropower generator. The controller parameters of hydropower and the PV generator can be coordinated and designed based on the unified frequency model.  
Robust Design of ULFO Suppression Control in Hydro-PV Coordinated Systems Based on the Structural Singular Value Method