## 4 Results  
The applied approach has been validated through simulation on the IEEE 39 bus test system [39]. Simulations are conducted on the IEEE 39 bus test system modeled in DIG SILENT Power Factory 2021 SP1 software (Gomaringen, Germany). It has been established that the generating buses are equipped with PMUs. In the Section 4, the first application of the MEMD method for estimating RoCoF will be presented, confirmed through references [25,38]. Figure 2 is designed to illustrate the dynamic response of operating generators following a specific disturbance scenario. The scenario entailed a three-phase short circuit occurring at bus 38 at a time instant of 0 s. This fault was subsequently cleared within 0.2 s. Concurrent with the fault clearance at 0.2 s, two generating units, namely generators G9 and G10, experienced an outage. The graph in Figure 2 specifically focuses on depicting the speed deviation in Hz for the remaining operational generators in the aftermath of this disturbance. The presented speed deviation values are in terms of (Hz), following the imposed disturbance of 0.108 (p.u.). A disturbance in the power system can be caused by various factors, such as sudden changes in load demand, equipment failures, or faults in the transmission or distribution network. These disturbances can lead to imbalances between power production and consumption, resulting in fluctuations in the system frequency. This figure is crucial for understanding the impact of the simulated short circuit and the subsequent generator outages on the stability and performance of the power system. The signals depicted in Figure 2 underwent the MEMD algorithm for computing the CoI RoCoF. This research was explicitly focused on representing \(\omega_{CoI}(t)\) as \(r_{n}(t)\), delineating the residual resulting from signal decomposition using the MEMD methodology. Figure 3 showcases these residuals, the remains after implementing the MEMD approach on the signals depicted in Figure 2.  
By employing the MEMD algorithm, it is capable of generating an average pattern and average residuals from the collective set of signals. This approach facilitates the simultaneous analysis of multiple signals, thereby yielding a unified solution that encompasses all individual signals and their respective residuals. Figure 3: Residuals of speed deviation obtained via MEMD. Figure 2: Speed deviation following disturbance. This characteristic can be utilized to obtain an averaged residual. In Figure 3, one can observe the curve depicted in black, representing the average modal function and residue across the complete signal set, which is utilized in this study to obtain CoI RoCoF. The gradient calculation or backward derivative was applied to this time series residual CoI using a calculation period of 0.01 s within an observed time of 2.5 s in PowerFactory (\((((x_{(i+1)}-x_{i}))/\Delta t)\Delta t=0.01\) s). To confirm the accuracy of the applied approach for estimating CoI RoCoF, the calculation of the CoI RoCoF involves the known total system inertia and the imbalance of active power in the system following a disturbance, using Equation (18):  
\[RoCoF=(f_{0}\times\Delta p)/(2\times H)=(60\times 0.108)/(2\times 6.93)=-0.467 \ (\text{Hz}/\text{s}).\]  
The formula for \(H=\frac{1}{S_{sys}}\sum_{i=1}^{N}H_{i}\cdot S_{i}=7.6969\ (\text{s})\) can be restated as follows. The reciprocal of the system's total apparent power \(S_{sys}=\sum_{i=1}^{N}S_{i}\) equals the sum of individual machine inertia constants \(H_{i}\) multiplied by their respective rated apparent power \(S_{i}\). \(S_{sys}\) refers to the equivalent system's apparent power, expressed in [MVA]; \(H\) is the system's inertia constant, measured in [s]; and \(S_{i}\) and \(H_{i}\) are the rated apparent power and the inertia constant of the \(i\)th generator. Additionally, there are \(N\) generators in the system. Furthermore, 1080 [MW] denotes the generation power deficiency \(\Delta p\), and \(f_{0}\) represents the nominal frequency in [Hz].