## 2 Methods for Inertia Estimation  
To address frequency stability issues, some works have been proposed to estimate the inertia in power systems with the purpose to improve the frequency stability and enable higher penetration of wind power plants [14; 20].  
In [20], a virtual inertia was estimated by using the frequency deviation measurements in a microgrid. The method was based on the swing equation, and the value of the inertia was used for tuning load frequency controllers of the generation units and calculating the active power required for an energy system storage, composed by Ultracapacitors (UC), to improve the frequency stability problems.  
The work in [21] calculated a synthetic inertia by means of a transfer function that represented the electromechanical behavior of a synchronous generator. The inertia estimation was done considering load changes and frequency deviations read of the power system.  
In the work reported in [22], a virtual inertia was estimated through the swing equation of synchronous machines. The system was linearized with respect to the speed of the generator, and the inertia was determined by the balance between the mechanical and active power of the generation units.  
The work in [23] estimated the power system inertia through a simulation model and frequency measurements from PMUs. In the inertia estimation, the frequency and the total active power at the connection point just after the disturbance were considered.  
In [24], the inertia estimation was based on an approximation of the total power change after a disturbance has occurred. This methodology used the average frequency of the system and the first instant after the application of the disturbance to estimate the inertia.  
The aforementioned methodologies were based on the swing equation and transients events, like the ROCOF, and some of them used off-line measurements, such as the power system frequency, which led to a low inertia estimation accuracy [25]. On the other hand, the method presented in [24] provided low inertia estimation error by employing particle swarm optimization, but the accuracy of this method depended on the load power change model and a large number of parameters such as currents, load impedance, power change measurements after the disturbance, and the frequency average of the system.  
Table 1 shows the differences between the main methodologies proposed in the literature for the inertia estimation and the methodology used in this research. The plus and minus signs indicate if the methodology contains (+) or not (\(-\)) the characteristic evaluated in Table 1, respectively. The principal characteristics used in the comparison are the simplicity of the algorithm, online implementation, accuracy of the inertia estimation on the first oscillation, mean error, a smaller number of parameters used for the estimation, and the robustness of the algorithm against variations in the system.  
In contrast with the the mentioned methods, the methodology used in this paper only needs the bus angle deviation at generation buses obtained from PMUs and real power measurements. It has a considerable balance of accuracy on the first oscillation, a simplified algorithm, robustness in the presence of variations in the system, and low error in the estimation of the inertia in the generation units. However, a disadvantage of this methodology is that the accuracy is reduced when the system is subject to small perturbations such as load variations. The methodology uses the time when the perturbation starts to determinate the peak value of the first angular oscillation. In its current form, this methodology does not include any technique for the detection of perturbations.