## 3 Materials and Methods
### RoCoF Estimation Using MEMD  
The replacement of traditional generation with inverter-based resources is gradually taking place. This makes the lack of rotational inertia common in modern power systems. This results in an increasing RoCoF after contingencies, potentially leading to frequency collapse. Precision in estimating RoCoF is vital for creating sophisticated operations and control methods in modern power systems, as RoCoF is a key indicator of the frequency, safety, and stability of these systems. Various approaches and papers describing RoCoF estimation methods can be found in the literature. In this paper, the MEMD approach is used to estimate RoCoF, the mathematical properties of which are presented in the previous chapter as well as in the references [25; 37]. Following a disruption in the power system that causes an imbalance between generation and consumption, generators either decelerate or accelerate to adjust to the new situation. Despite individual generators slowing down at different rates (\(d\Delta\omega_{i}/dt\)), which change throughout the process, the system as a whole slows down at a constant rate (\(d\Delta\omega_{c}/dt\)). As synchronized forces act, damping the oscillation in the stable response, the frequencies of individual generators approach the frequency of the center of inertia. After a relatively short time (\(t=t_{1}\)), of the order of a few seconds, all generators practically adopt the frequency values of the center of inertia, thereby reestablishing a unified frequency within the system. This feature of the power system allows the application of an advanced MEMD approach signal processing technique to be applied and used to assess the trend of system deceleration or acceleration in a newly emerging situation after a disturbance. In this paper, it is shown through an illustrative example that by using the technique of empirical decomposition of the MEMD signal over frequency signals, it is possible to estimate the value of the rate of change of frequency and thus the value of the rate of change of the center of inertia frequency with high accuracy. The first step requires access to the frequency signal from WAMS (PMU). Subsequently, the MEMD algorithm segregates these signals into several IMFs and residuals. The residuals, generated through the MEMD method, effectively capture the signal's trend. It is demonstrated that these residuals closely align with the calculated center of inertia, making them well suited for estimating \(df_{Col}/dt\). The extracted IMF is removed from the original signal \(f_{i}(t)\), leading to the calculation of the signal residual for subsequent IMF functions: \(r_{1}=f_{i}(t)-c_{1}\). The aggregate of all IMFs along with the residual constitutes the initial signal:  
\[f_{i}(t)=\left(\sum_{j=1}^{n}c_{j}(t)\right)+r_{n}(t) \tag{13}\]  
where \(f_{i}(t)\) signifies the generator frequency, \(c_{j}(t)\) stands for IMFs, \(r_{n}(t)\) indicates a residual or a low-frequency trend, and n represents the overall count of IMFs. In this context, research is specifically concentrated on accurately representing \(\omega_{Col}(t)\) using \(r_{n}(t)\) and examining its suitability for use in the evaluation of the center of inertia estimation. Utilizing various iterations of EMD per signal, MEMD can produce the average modal function and residue across the complete signal set. This enables the concurrent processing of multiple signals, establishing a unified resolution encompassing all signals and their residual elements. The simultaneous decomposition of all measured data from PMU happens with MEMD. It identifies scales shared among two or more signals. This unique alignment ability allows in-depth analysis across multiple signals. For each variable, MEMD offers IMFs and residuals separately. The combination of all modes and the leftover residual mirrors the initial signal. Finding local extremities within MEMD employs real-valued signal projections to form a multidimensional envelope.