## 1 Introduction  
The amount of wind power generation has grown in recent years, providing 3% of global power, for example, China is leading with 164 GW, the United States with 6 GW, Germany with 6 GW, India with 4 GW, and the U.K. with 4 GW [1]. This is along with the aim of reducing the dependence on fossil fuels and mitigating the adverse effects of global warming [2, 3]. Wind generation brings benefits [4, 5], but also challenges for power systems [6, 7], especially when there is a high penetration of wind power plants. Therefore, it is essential to conduct studies about the stability and operation of power systems, including this type of plant. The determination of the stability of the power system is important; it lets us know if the power system is able to regain a state of equilibrium after the system is subjected to a physical disturbance [8]. Inertia is the parameter that represents the capability of rotating machines to store and inject their kinetic energy in the system. The level of inertia influences the Rate Of Change of Frequency (ROCOF) and transient frequency values during a system perturbation. The transient value of the system frequency is important because variations in the frequency caused by an inertia decrease can raise the risk of reaching values that are dangerous for the system frequency stability (generator trip, load shedding intervention, etc.) [9]. On the other hand, wind turbine generators are decoupled from the grid by power electronic converters; thus, they do not directly contribute to system inertia [10], and large-scale penetration of non-inertial wind generators in the power system will result in reduced system inertia [11; 12; 13]. This reduction has a substantial impact on frequency regulation [14]. Therefore, the inertia estimation can be used as an input for the control of power electronic converters and energy storage systems, among others, helping to maintain the system frequency stability during disturbances. In the literature, potential applications have been proposed for the results obtained from the inertia estimation. For instance, in [15], the inertia estimation was used to enhance the dynamic operation and performance of a power grid against frequency stability problems, while in [16], an inertia estimation algorithm was used in combination with the ROCOF and a derivative control of a High Voltage Direct Current (HVDC) controller to provide active power and enhance the inertia response of conventional generators when the power demand changes. However, the main application of the inertia estimation in wind power plants is to set the references of the synthetic inertial controllers included in these plants [17; 18]; these controllers are based on the concept of hidden inertia emulation of wind power plants, which increases the electric power output during the initial stages of stability frequency issues using the kinetic energy from moving parts. The concept of synthetic inertia provides a boost energy in a similar way as a synchronous generator with real inertia and reduces stability frequency issues. This paper improves and evaluates the methodology presented in [11] for the estimation of the inertia available in hydro and wind power plants based on the measurements obtained from PMUs. The methodology was modified using a fourth-order finite-difference approximation to calculate the second-order derivative of the swing equation, and the results obtained are way better. Besides, the methodology is tested using measurements obtained from industrial PMUs in a test power system implemented in an RTDS, to test its practical application, and not only using offline simulations, as reported previously. In addition, the methodology is applied to an equivalent of the Mexican southeast power system, where there is high penetration of wind and hydropower generation.