## 4 Results
The results for the gradient calculation or backward derivative applied to the time series residuals of frequency deviation from generator terminals and residual CoI are presented in Table 1. These calculations were performed using a calculation period of 0.01 s within an observed time of 2.5 s. In addition, the DigSILENT PowerFactory provides the rate of change of frequency. It calculates the RoCoF utilizing the backward derivative within the PowerFactory, employing the same calculation period and observed time duration for the assessment. In addition, the outcomes from the MEMD analysis and the computation of \(d\omega_{CoI}/dt\) via DigSILENT were compared with the results generated by Equation (18). In order to statistically process the results, the mean square and mean absolute errors are calculated. Table 2 shows the results for mean squared error and mean absolute error. \(d\omega_{CoI}/dt\) was set to \(-0.467\ (\text{Hz}/\text{s})\). \[MSE(d\omega/dt)=\frac{\sum_{i=1}^{N}\biggl{(}d\overset{\sim}{\omega_{i}}/dt-d \omega_{CoI}/dt\biggr{)}^{2}}{N} \tag{19}\]  
\[MAE(d\omega/dt)=\frac{\sum_{i=1}^{N}\biggl{|}d\overset{\sim}{\omega_{i}}/dt-d \omega_{CoI}/dt\biggr{|}}{N} \tag{20}\] \begin{table}
\begin{tabular}{c c c} \hline \hline  & **MEMD** & **DigSILENT** \\ \hline \(d\omega_{1}/dt\) [Hz/s] & \(-0.43959\) & \(-0.42978\) \\ \(d\omega_{2}/dt\) [Hz/s] & \(-0.45777\) & \(-0.44485\) \\ \(d\omega_{3}/dt\) [Hz/s] & \(-0.44931\) & \(-0.45509\) \\ \(d\omega_{4}/dt\) [Hz/s] & \(-0.44713\) & \(-0.46232\) \\ \(d\omega_{5}/dt\) [Hz/s] & \(-0.45623\) & \(-0.46274\) \\ \(d\omega_{6}/dt\) [Hz/s] & \(-0.45834\) & \(-0.45025\) \\ \(d\omega_{7}/dt\) [Hz/s] & \(-0.45918\) & \(-0.45868\) \\ \(d\omega_{8}/dt\) [Hz/s] & \(-0.45961\) & \(-0.45756\) \\ \(d\omega_{CoI}/dt\) [Hz/s] & \(-0.4534\) & \(-0.45266\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The results of the evaluation of the rate of change of speed deviation using the MEMD method. The error analysis conducted in this paper, encompassing mean square error and mean absolute error, offers fundamental insights into methodological accuracy. Nevertheless, the practical implications of these errors in power system operation and stability warrant further exploration. In practical scenarios, inaccuracies in inertia estimation could significantly impact the system's frequency response and stability, leading to challenges in frequency control and potentially escalating to broader system instabilities. Future research should focus on detailed simulations and case studies to quantify the impacts of error margins on operational strategies, stability margins, and grid reliability. Within the scientific research, an analysis is conducted on the percentage error between the calculated values of \(d\omega_{Col}/dt\) using the MEMD method and calculations in the DigSILENT PowerFactory 2021 SP1 software in comparison to the actual computed value of \(d\omega_{Col}/dt\), as expressed in Equation (18):  
\[error=\left(\frac{d\omega_{1}^{\ \sim}/dt-d\omega_{Col}/dt}{d\omega_{Col}/dt} \right).100\% \tag{21}\]  
The result of the estimation error (%) are shown in Table 3.  
To validate the proposed approach for estimating the inertia of the electric power system, two cases were analyzed. The first case pertains to a system with dominant production using synchronous generators. Three experiments were simulated, comprising one medium and two large disturbance scenarios. For the medium scenario, an active power imbalance of 0.05 p.u. was set, while for large disturbances, active power generation-consumption differences of 0.1 (p.u.) and 0.108 (p.u.) were, respectively defined. The second case concerned the estimation of the electric power system's inertia on a modified IEEE 39 bus test system with a share of electricity production from renewable energy sources (RESs). In addition, disturbances were simulated as in the previous case, but the analyses were conducted with a share of RES generation amounting to 10%, 15%, and 20%, respectively.