## 3 Methodology for the Inertia Estimation  
Typically, the natural inertia of synchronous generator units responds voluntarily when the system is subject to disturbances, without any control actions [26; 27]. This behavior can be achieved in wind power plants through power electronic converters by wind turbine inertia controls, even when they are decoupled partially or completely from the grid [12; 28; 29]. This assumption is valid because modern wind turbine controllers can convert the kinetic energy into active power, i.e., control functions achieved by power converter controls can be seen as the virtual inertia of wind generators [14; 18]. This is shown in Figure 1. Through both natural and virtual inertias, active power is released to the power system from the kinetic energy of generation units. In the case of wind power plants, this is done through power electronics converters and wind turbine controls that extract the kinetic energy from moving parts and convert it into real power output [17; 18]. This characteristic of inertia response in wind power plants can be described in a format similar to conventional generator units by the swing equation, which is directly derived from [PERSON]'s law of motion on rotating objects. Due to this correlation between the virtual inertia of wind power plants and the real inertia of synchronous generation units, an inertia can be estimated for wind power plants in a format similar to conventional generators [11]. The characteristic inertia response can be represented by the swing equation, which describes the change between the axis of the rotor and the magnetic field. This change is produced when a disturbance occurs in the power system. As a result, the rotor accelerates or decelerates with respect to system synchronous speed [26]. The swing equation is given by the next expression [27; 30]:  
\[\frac{2H}{\omega_{0}}\frac{\partial^{2}\Delta\delta}{\partial t^{2}}=\Delta P_ {M}-\Delta P_{E}-K_{D}\Delta\omega \tag{1}\]  
where \(\omega_{0}\) is the rated speed of the system, \(\delta\) is the rotor angle, \(P_{M}\) is the mechanical power, \(P_{E}\) is the electrical power, \(K_{D}\) is the damping factor, and \(\Delta\omega\) is the derivative of the rotor angle. Equation (1) represents the dynamic response of the inertia of the synchronous machines or a group of synchronous machines. Rotor angle is the rate of change of the angular displacement of the rotor, that is the angular speed of the rotor is equivalent to the derivative of the angle of the rotor with respect to time. This expression can be represented as [11]:  
\[\Delta\omega=\frac{\partial\Delta\delta}{\partial t} \tag{2}\]  
The rotor angle \(\delta\) cannot be directly measured; however, the angle of the voltage on the bus \(\theta\) closely follows the rotor angle. Thus, Equation (1) can be rewritten as [11]:  
\[\frac{2H}{\omega_{0}}\frac{\partial^{2}\Delta\theta}{\partial t^{2}}=\Delta P _{M}-\Delta P_{E}-K_{D}\frac{\partial\Delta\theta}{\partial t} \tag{3}\]  
Real-time measurements, like mechanical power, could be difficult to obtain, especially when there is a change in the mechanical power at the turbine of each generation unit. A consideration can be applied to Equation (3) assuming that the mechanical power has a very large time constant compared with the electrical power. During the first swing of any disturbance, when primary frequency controls are not yet active, it is valid to assume that the mechanical power output remains constant. In addition, the mentioned consideration for this methodology is applied for short-term transient events where the mechanical torque has a higher value in comparison with electromagnetic torque [27; 31]. Assume:  
\[\Delta P_{M}=0\]