## 2 Transfer Functions of Components in the Frequency Control System  
This paper analyzes the frequency stability control in multi-area power systems, with a particular emphasis on hydropower plants. Simulation scenarios are carried out for cases where hydropower plants are combined with other types of power plants. To support modeling and simulation, the components of the power system are represented using transfer function equations. These equations play a crucial role in building mathematical models to describe the detailed operation of the system. This makes it possible to examine how the system reacts in various scenarios and create efficient control strategies to preserve peak performance. Hydropower plants stand out for their ability to quickly adjust their output to handle sudden load fluctuations by adjusting water turbine settings. This makes them an ideal option for handling unpredictable surges in demand. On the other hand, other types of power plants, particularly thermal power plants, often have slower response times. Therefore, studying the combination of hydropower plants with other power plant types can help us to identify effective solutions to stabilize frequency and synchronize the operation of the power system. The hydro governor is an essential component in regulating water flow to the turbine based on the control signals provided by the PID controller. Its dynamic behavior can be represented by the following transfer function [33]:  
\[G_{gov}(s)=\frac{1}{T_{gh}s+1}\frac{T_{rs}s+1}{T_{rh}s+1} \tag{1}\]  
where \(T_{gh}\), \(T_{rs}\), and \(T_{rh}\) are time constants that characterize the dynamic behavior of the hydro governor system. The water turbine converts energy from water flow into mechanical power supplied to the power system. Its operation is described by the following transfer function [33]:  
\[G_{turbine}(s)=\frac{-T_{w}s+1}{0.5T_{w}s+1} \tag{2}\]where \(T_{w}\) is the penstock time constant which characterizes the effect of water inertia in the penstock on the system's response. The speed governor monitors the frequency deviation between the actual frequency and the nominal value (50 Hz) to control the steam flow to the turbine. This process helps maintain the generator's speed and power output. The transfer function of the speed governor is represented as follows [33]:  
\[G_{gou}(s)=\frac{1}{T_{sg}s+1} \tag{3}\]  
where \(T_{sg}\) is the speed governor time constant, representing the response time of the speed governor in regulating the turbine's output power. The non-reheat turbine directly converts the thermal energy of steam into mechanical energy to drive the generator. This type of turbine operates with a simple configuration, and it can be described using the following transfer function [33]:  
\[G_{turbine}(s)=\frac{1}{T_{t}s+1} \tag{4}\]  
where \(T_{t}\) is the turbine time constant, representing the inherent delay in the turbine's power output response to changes in the control input. The reheat turbine, unlike the non-reheat type, has an additional reheating stage in the steam cycle. After passing through the high-pressure stage, the steam is heated to a higher temperature before proceeding to the low-pressure stage. This reheating process improves the turbine's thermal efficiency by increasing the energy content of the steam, which allows for better performance in later phases. Because of this stage, the transfer function of the reheat turbine is more complicated and is represented as follows [33]:  
\[G_{reheat}(s)=\frac{K_{r}T_{r}s+1}{T_{r}s+1} \tag{5}\]  
where \(K_{r}\) is the reheat gain, representing the ratio of reheated turbine power to total turbine power, while \(T_{r}\) is the reheat time constant, representing the delay associated with the reheating process in the turbine. The power system converts mechanical energy from the turbine into electrical energy for the loads. It also generates the frequency deviation signal by comparing mechanical and consumption power, serving as the foundation for frequency control.