Characterization of Power System Oscillation Modes Using Synchrophasor Data and a Modified Variational Decomposition Mode Algorithm

Abstract

The growing complexity and uncertainty in modern power systems—driven by increased integration of renewable energy sources and variable loads—underscore the need for robust tools to assess dynamic stability. This paper presents an enhanced methodology for modal analysis that combines Adaptive Variational Mode Decomposition (A-VMD) with Prony’s method. A novel energy-based selection mechanism is introduced to determine the optimal number of intrinsic mode functions (IMFs), improving the decomposition’s adaptability and precision. The resulting modes are analyzed to estimate modal frequencies and damping ratios. Validation is conducted using both synthetic datasets and real synchrophasor measurements from Ecuador’s national power grid under ambient and disturbed operating conditions. The proposed approach is benchmarked against established techniques, including a matrix pencil, conventional VMD-Prony, and commercial tools such as WAProtector and DIgSILENT PowerFactory. The results demonstrate that A-VMD consistently delivers more accurate and robust performance, especially for low signal-to-noise ratios and low-energy ambient conditions. These findings highlight the method’s potential for real-time oscillation mode identification and small-signal stability monitoring in wide-area power systems.

1. Introduction

In modern power systems, operational and dynamic challenges have steadily increased due to the integration of renewable energy sources and the growing use of powered electronic devices. These evolving operating conditions have reduced the system’s mechanical inertia, giving rise to stability issues associated with electromechanical oscillations. Such oscillations, inherently present during regular grid operation, are typically quasi-stationary due to the continuous balance between energy production and consumption through transmission lines. Moreover, as interconnected power networks often operate close to their capacity limits, the system’s overall stability is further compromised. If not correctly damped, these oscillations can threaten system stability and significantly constrain power transfer capabilities, potentially leading to widespread system collapse [1,2]. This situation is particularly relevant in electric grids with a high share of run-of-the-river hydroelectric plants, where the limited regulation capacity of these facilities restricts their ability to respond to short-term fluctuations in supply and demand, increasing the risk of operational instability during periods of low water availability [3].

Monitoring the dynamic behavior of Power Electric Systems (PESs) using phasors derived from sinusoidal voltage and current signals at system buses has emerged as a practical approach to assess system stability. By properly analyzing the oscillatory behavior of these operating variables, valuable insights can be gained into system dynamics. These sinusoidal signals, which reflect the dynamic state of the power grid, are captured through Phasor Measurement Unit (PMU) technology, capable of estimating phasors at rates between 10 and 60 samples per second [4]. The data collected by Phasor Measurement Units (PMUs), combined with monitoring, protection, and control applications, constitute a wide-area measurement system (WAMS) [5].

WAMS integrates several technological components that process the dynamic operation data of the electrical system, including communication networks, signal processing techniques, and specialized monitoring applications tailored to each system developer. Together, these elements enable real-time visualization of PES dynamics, primarily through frequency and power measurements [6]. Due to PMUs’ ability to capture high-resolution data on the system’s instantaneous state, large volumes of information are generated for monitoring. Depending on the accuracy and efficiency of the applications integrated into WAMS, it is possible to estimate in real time the characteristics of system oscillations, such as frequency and damping of oscillation modes, with high update rates (at least one cycle) and minimal delays [7,8,9].

Traditional engineering approaches for identifying electromechanical oscillation modes have relied on model-based methods, which process a limited set of operational signals. These methods typically involve linearizing the system around an equilibrium point, enabling system dynamics analysis through eigenvalues and eigenvectors. While effective for understanding system behavior, model-based techniques require frequent updates to account for changing operating conditions, limiting their practicality.

The advancement of real-time monitoring technologies has introduced measurement-based approaches that leverage continuous data acquisition from PMUs. Unlike model-based methods, PMUs provide precise, time-synchronized measurements from geographically distributed buses using the Global Positioning System (GPS). This real-time data integration, facilitated by WAMS, enhances the detection of poorly damped oscillations and enables large-scale visualization of system dynamics. Consequently, WAMS supports comprehensive stability assessments, allowing operators to make informed decisions based on current operating conditions [10].

Furthermore, PMUs have significantly improved power system oscillation analysis by enabling the application of advanced signal processing techniques. Several methods have been proposed for the identification of electromechanical oscillation modes in power systems. Classical approaches such as Prony analysis [11], autoregressive moving average (ARMA) models [12], wavelet transforms [13], and matrix pencil methods [14] have been extensively studied and applied in various scenarios. While these techniques can be effective under controlled or simulated conditions, they often face significant limitations when applied to non-stationary or noisy synchrophasor signals, particularly in ambient operating conditions. More recently, methods based on advanced signal decomposition, such as Empirical Mode Decomposition with the Hilbert–Huang Transform (EMD-HHT) [15], Singular Spectrum Analysis (SSA) [16], and improved adaptive VMD [17], have emerged to overcome some of these challenges. However, many of these approaches lack real-time feasibility or require manual tuning of decomposition parameters, which can limit their practical deployment in wide-area monitoring systems.

These techniques make it possible to identify dominant oscillatory modes, estimating their frequency and damping ratios and differentiating between local and inter-area oscillations. Additionally, integrating PMU data with these methods enhances the robustness of modal analysis by improving noise resilience and enabling real-time tracking of system dynamics.

Research using WAMS information has focused on ensuring the proper functioning of the various calculation and signal analysis algorithms. For example, high precision is assumed for the algorithms processing the voltage and current positive-sequence variables obtained through the PMUs, with which the system’s operation is monitored in terms of frequency, active power, and reactive power [12,18]. Other signal analysis algorithms can detect and locate disturbances in the system, as proposed in [19], and identify poorly damped oscillation modes, as presented in [14]. These studies assume that the WAMS information is clearly defined during the disturbance period, as are data obtained from PES simulation software.

However, signals recorded in real systems exhibit permanent variability related to the inherent dynamics of the power system. Hence, even during static periods, the PES is in a quasi-stationary state, and so signal analysis algorithms (including those in commercial applications) are prone to estimation errors.

During regular system operation, oscillations are triggered by various sources, reflecting the quasi-stationary behavior of the system, with fluctuations often referred to as “ambient” [20]. The primary excitation sources in these environmental conditions stem from the combined effects of variable loads, minor transients, small generation changes, and occasional faults, all of which contribute to environmental excitation for electromechanical oscillations. These signals contain valuable information about the system’s operational state and can be leveraged for early detection of oscillations. However, ambient noise often masks these environmental signals, making it challenging to extracting meaningful data that accurately characterize the system’s quasi-stationary operation. This challenge remains the subject of ongoing research [15].

PES control centers equipped with wide-area measurement system (WAMS) technology rely on commercial software to analyze system operation and stability. In Ecuador, for example, the National Electricity Control Center (CENACE), an Independent System Operator (ISO), employs WAProtector, a solution developed by the Slovenian company ELPROS [21]. This software collects real-time phasor data from PMUs and performs advanced data analysis to monitor grid dynamics. Its applications include the following [22]:

• Static angle stability;
• Voltage stability of transmission corridors;
• Island detection;
• Harmonic information of the system;
• Oscillatory stability;
• System events;
• Historical data analysis.

Through the WAProtector applications, operators can perform specialized analyses using data collected from each PMU, properly synchronized to provide valuable insights into various phenomena occurring within the system. One of these applications enables real-time monitoring of the system’s oscillatory stability, primarily focusing on identifying the oscillation modes present in the analyzed signal through a proprietary modal estimation algorithm. As commercial software, the different algorithms within the application are accompanied only by a user manual [23]. The output of the WAProtector modal estimation algorithm provides key variables for determining the oscillatory stability level, such as the amplitude (MW), damping rate (%), and frequency (Hz) of the oscillation modes present in the signal. Although this application also makes it possible to set average values for each characteristic variable of the resulting oscillatory stability at specific time intervals (with a minimum window of 4 min) [13,24], these characteristics can exhibit significant variability.

In this context, the averaged signals may not be suitable for other applications or stability monitoring by the electrical operator, as they may contain outliers or NaN values. Since PMUs, like any other equipment, have inherent accuracy limitations and can experience computational errors due to the system’s dynamics, WAMS applications rely on information derived from the actual operation of the system, which is affected by various transients, such as peaks, drops, frequency variations, and DC components. These effects, though minimized, can still distort the signals. Therefore, real-time signals used to identify oscillation modes based on the synchrophasor measurements variables and mathematical algorithms are susceptible to such distortions. This underscores the necessity for additional analysis to obtain reliable information regarding oscillatory behavior.

The analysis of oscillations in power systems remains an active area of research. Although significant progress has been made, there is still a lack of preprocessing algorithms capable of detecting signals with oscillatory modes before further analysis. Incorporating such algorithms could reduce computational complexity by discarding signals without oscillatory modes. Additionally, optimizing existing approaches is essential to address the challenges of larger-scale, interconnected systems [17,23,25]. Valuable dynamic information can be extracted by identifying and analyzing the system’s output signal.

WAMS has recently gained widespread application in energy systems, providing baseline signals for signal analysis techniques [6]. One notable technique, VMD, has shown promising results when applied to environmental and oscillatory power signals from a power system generator [17]. VMD has also been combined with algorithms such as Prony’s method to enhance the accuracy with which low-frequency oscillation modes are identified. This combined approach helps eliminate interference in the original signals, allowing a more precise determination of modal parameters. However, the practical implementation of VMD requires considerable manual intervention to determine the number of modes and adjust penalty parameters, which limits its efficiency in real-world applications [17,25].

In recent years, increased attention has been directed toward analyzing and mitigating low-frequency and wide-band oscillations in power systems with high penetration of renewable energy sources. For instance, Ref. [26] offers a comprehensive review of the mechanisms driving wide-band oscillations in inverter-based renewable systems, emphasizing the challenges traditional model-based techniques face in such fast-changing and weakly damped environments. In response to these limitations, the authors of [27] developed a hybrid approach that integrates Improved Variational Mode Decomposition (IVMD) with sparse time-domain analysis, demonstrating enhanced robustness against noise and better adaptability under dynamic system conditions. Finally, the work in [28] further advanced the field by proposing a real-time PMU-based method for oscillation detection and mitigation in smart grids, improving situational awareness and control responsiveness through synchrophasor data analytics. Collectively, these recent contributions highlight a growing shift toward adaptive, data-driven methods—reinforcing the importance of advancing such techniques for reliable oscillation mode identification in modern power systems.

Although the use of VMD, often in combination with Prony analysis, has been reported in previous studies for oscillation mode identification in power systems [17,25,29], most existing implementations rely on manually defined parameters, such as the number of modes (K), limiting their effectiveness in real-time or automated contexts. In contrast, this work introduces an enhanced A-VMD methodology that autonomously selects K based on energy concentration across predefined frequency bands. This adaptivity improves the decomposition of both ambient and transient signals.

Moreover, much of the existing literature on oscillation mode identification using VMD-based or similar techniques has been limited to simulation environments or standard benchmark systems. There is a noticeable gap concerning the validation of these data-driven approaches using real synchrophasor measurements from operational power systems, particularly under varying conditions that include both ambient fluctuations and disturbance events. This lack of practical evaluation raises questions about the generalizability and reliability of such methods for deployment in real-world WAMS.

To benchmark the robustness and operational relevance of the proposed methodology, we include a comparative analysis with both commercial tools (WAProtector, DIgSILENT PowerFactory) and recent academic techniques.

Table 1. Comparison of the proposed A-VMD method with commercial and recent academic techniques.

Reference	Techniques						Commercial Use
	Noise Robustness	Ambient Signal Analysis	Real-Time Suitability	Manual Tuning Required	Modal Resolution	Adaptive Mode Selection
Prony Analysis [11]	No	No	Yes	Yes	Yes	No	Yes
ARMA Models [12]	No	No	Yes	Yes	Yes	No	No
Wavelet Transform [13]	Yes	No	Yes	Yes	No	No	No
Matrix Pencil [14]	Yes	No	Yes	Yes	Yes	No	Limited
EMD-HHT [15]	No	Yes	No	No	Variable	No	No
SSA [16]	Yes	Yes	No	No	Yes	No	No
IA-VMD [17]	Yes	Yes	No	Yes	Yes	Yes	No
WAProtector [11]	Moderate	No	Yes	Yes	Yes	No	Yes
DIgSILENT PowerFactory [11]	Moderate	No	Yes	Yes	Yes	No	Yes
A-VMD	Yes	Yes	Yes	No	Yes	Yes	No

summarizes key features such as noise robustness, real-time suitability, modal resolution, and adaptive capabilities, underscoring the advantages of A-VMD in dynamic and low-inertia environments.

A specialized literature review shows that low-frequency oscillations present a critical issue in modern electrical systems. They can undermine power grids’ stability and safe operation if improperly damped. Therefore, developing advanced analysis tools based on measurements and signal processing techniques, such as VMD, offers promising solutions. However, further improvements are necessary for their practical application in the dynamic and complex scenarios presented by contemporary electrical systems.

Thus, this study aims to address these knowledge gaps. Specifically, in this work, we found the level of oscillatory stability of PES by determining the critical oscillatory modes by processing synchrophasor measurements of oscillation obtained from PMUs, along with ambient data; these data were analyzed using the combined VMD and Prony technique so that ISO could be used to obtain the critical oscillation modes, which, with a commercial system, may present errors due to the need to adjust the parameters to the operating conditions properly. The main contributions of this study can be summarized as follows:

• Development of a mode identifier for an electrical system using synchrophasor measurements through the VMD algorithm;
• An extensive comparison between the commercial technique WAProtector, modal analysis with DIgSILENT PowerFactory, and the matrix pencil technique;
• Concrete recommendations on the determination of oscillation modes through oscillatory and ambient synchrophasor measurements, which can help to further develop more efficient damping of power oscillations in the future.

The remainder of this paper is structured as follows: Section 2 presents the theoretical framework for the analysis conducted in this study. Section 3 explores the application of the proposed methodology to the Ecuadorian electricity system. Section 4 discusses the advantages of the proposed method. Finally, Section 5 concludes the paper, highlighting key findings and suggesting directions for future research.

2. Theoretical Framework

This section introduces the VMD technique, which is used in the following section as a key tool for identifying dominant oscillation modes in synchrophasor measurement data, including both electromechanical oscillations and environmental disturbances. Subsequently, the concept of intrinsic mode functions (IMFs) is introduced, followed by a discussion on estimating the optimal number of modes.

2.1. Variational Mode Decomposition

The VMD method decomposes a signal into a finite set of components known as IMFs, each representing a distinct frequency mode extracted from the original signal. Unlike the Empirical Mode Decomposition (EMD) method, which relies on an iterative sifting process, VMD formulates the decomposition as an optimization problem, treating it as an adaptive Wiener filter bank. This approach enhances decomposition stability, reduces mode mixing, and provides improved frequency resolution [29].

VMD has emerged as a powerful signal analysis tool, particularly in applications requiring precise mode separation and noise robustness. Its ability to adaptively extract meaningful oscillatory components makes it well-suited for analyzing synchrophasor data in power systems, facilitating the identification of dominant electromechanical oscillation modes and environmental disturbances.

The original signal generally comprises environmental oscillations at various frequencies, along with trend components and noise introduced during measurement and transmission. Mathematically, this can be expressed as

$$s_0\left(t\right)=\sum _{i=1}^N{s}_i\left(t\right)+R\left(t\right)+n_0\left(t\right),$$

(1)

where the variables are defined as follows:

• $s_0\left(t\right)$: the measured signal obtained from the WAMS system;
• $s_i\left(t\right)$: the oscillatory component corresponding to the ith frequency of the signal;
• $R\left(t\right)$: the residual component containing the trend;
• $n_0\left(t\right)$: environmental and transmission noise present in the system.

The oscillatory component at the ith frequency of the signal is composed of individual IMFs, each capturing a distinct oscillatory behavior. These IMFs collectively represent the frequency-specific dynamics of the signal and can be expressed as [29]:

$$s_i\left(t\right)=\sum _{k=1}^K{u}_k\left(t\right),$$

(2)

where $u_k\left(t\right)$ denotes the k-th IMF of the inherent oscillation mode, given by:

$$u_k\left(t\right)=U_k\left(t\right)e^{-D_k\left(t\right)}cos\left(w_k\right),$$

(3)

where $U_k\left(t\right)$ represents the amplitude of the kth oscillatory component, $D_k$ is the damping ratio of the kth oscillatory component, and $w_k$ is its frequency.

For ambient signals obtained from power system operations, preprocessing steps are required to identify the trend, as well as remove missing data and outliers. Once the signal is processed, the trend is removed, and a Fast Fourier Transform (FFT) is applied to determine the number of frequencies K present in the ambient signal. Using this value of K and applying the VMD technique to the processed signal, the IMFs can be extracted [17].

Since the signal obtained from the WAMS system is a comprehensive monitor of system operations, it includes both ambient signals and identifiable oscillatory components. The model applied must be capable of extracting the oscillation modes present in the signal. The VMD technique serves this purpose, allowing for the extraction of frequency, amplitude, and damping parameters. These parameters can then be computed for each IMF component using the Prony algorithm.

VMD is recognized for its superior performance in identifying oscillation modes within non-stationary signals, especially in noisy conditions. It effectively mitigates aliasing, which occurs when the dominant frequencies of two IMFs are close to each other, ensures fast convergence, and exhibits greater robustness than the EMD technique. While EMD extracts IMFs from the signal envelope, VMD analyzes frequency bands to determine the central frequency of each IMF, similar to the analysis of AM-FM band signals [30].

2.2. Intrinsic Mode Functions

IMFs are obtained by decomposing the signal $s_i\left(t\right)$ into multiple components $u_k\left(t\right)$, as defined in (2). The VMD technique ensures that each component has a well-defined central frequency $w_k$ and a limited bandwidth. To achieve this, VMD employs a variational approach, where each mode $u_k\left(t\right)$ is iteratively refined to minimize interference between modes while preserving essential oscillatory characteristics.

The bandwidth of each component is estimated using the $H_1$ norm, which quantifies the Gaussian smoothness of the demodulated signal. This smoothness constraint plays a crucial role in the optimization process by enforcing spectral compactness, thereby preventing mode mixing and ensuring that each IMF remains localized in the frequency domain. The challenge in determining these components is formulated as follows [29]:

$$\begin{array}{cc}\hfill & \underset{\left\{u_k\right\},\left\{w_k\right\}}{min}\left\{\sum _{k=1}^K{∥\partial t\left[\left(\delta \left(t\right)+{\displaystyle \frac{j}{\pi t}}\right)\times u_k\left(t\right)\right]e^{-j{w}_k\left(t\right)}∥}^2\right\}\hfill \\ \hfill & \mathrm{subject}\mathrm{to}\sum _{k=1}^K{u}_k=s_i\left(t\right),\hfill \end{array}$$

(4)

where $\partial t$ represents the partial derivative with respect to time, and $\delta \left(t\right)$ denotes the unit impulse function. This formulation seeks to iteratively extract modes by minimizing the sum of their bandwidths in the spectral domain, thereby ensuring an optimal and well-separated signal decomposition.

To obtain the optimal solution to the VMD problem in (4), the method employs Lagrange multipliers $\lambda$ to convert the constrained optimization problem into an unconstrained one. This allows the problem to be tackled using variational calculus and iterative optimization techniques. The resulting augmented Lagrangian formulation is expressed as follows:  

$$\begin{array}{cc}\hfill L(\left\{u_k\right\},\left\{w_k\right\},\lambda )& =\alpha \sum _{k=1}^K{∥\partial t\left[\left(\delta \left(t\right)+{\displaystyle \frac{j}{\pi t}}\right)\times u_k\left(t\right)\right]e^{-j{w}_k\left(t\right)}∥}^2\hfill \\ \hfill & +{∥\sum _{k=1}^K\left(u_k-s_i\left(t\right)\right)∥}^2+\langle \lambda \left(t\right),\sum _{k=1}^K\left(u_k-s_i\left(t\right)\right)\rangle ,\hfill \end{array}$$

(5)

where the first term enforces spectral compactness by minimizing the bandwidth of each mode, the second term ensures that the sum of the extracted modes reconstructs the original signal as closely as possible, and the third term incorporates the constraint enforcement via the Lagrange multiplier $\lambda \left(t\right)$.

The term $\langle \lambda \left(t\right),{\sum }_{k=1}^K\left(u_k-s_i\left(t\right)\right)\rangle$ in (5) represents the inner product between the Lagrange multiplier $\lambda \left(t\right)$ and the reconstruction error, ensuring that deviations from the original signal are penalized. This enforces adherence to the constraint ${\sum }_{k=1}^K{u}_k=s_i\left(t\right)$, indicated in (4), during the optimization process. The optimization algorithm’s iterative update of $\lambda \left(t\right)$ progressively corrects deviations, guiding the solution towards a decomposition where the sum of the IMFs matches the original signal.

2.3. Estimation of the Number of Modes K

Since the value of K determines the number of modes to be extracted from the original signal, its determination directly impacts the accuracy of the decomposition. Selecting an inadequate K can lead to mode mixing (underestimation) or artificial mode generation (overestimation), affecting the interpretation of oscillatory components.

In the context of WAMS signals used for analyzing oscillation modes in stability studies, an additional filtering stage can be applied to the original signal to isolate specific frequency bands associated with inter-area, local, and higher-frequency modes. This pre-processing step enhances the adaptability of the VMD technique by focusing the decomposition on relevant oscillation bands, thereby facilitating a more accurate K estimation.

Several methods have been proposed in the literature for determining an optimal K. However, in practical implementations, an iterative approach is often adopted, where K is initially set based on prior knowledge or spectral analysis, followed by refinement through validation techniques such as residual energy assessment or mode orthogonality verification. By integrating these techniques, the VMD framework can be effectively adapted to the characteristics of synchrophasor measurements, ensuring robust mode decomposition for power system stability assessment.

3. Methodology and Problem Statement

Ecuador’s national power system exhibits several distinctive characteristics that make it a compelling case study for testing advanced modal analysis techniques. The country’s electricity generation is predominantly hydro-based, with a considerable proportion derived from run-of-river hydroelectric plants that offer limited storage and regulation capacity. This configuration increases the system’s sensitivity to short-term supply-demand imbalances, particularly during dry seasons when water availability is constrained. Furthermore, Ecuador’s National Interconnected System (SNI), coordinated by the independent system operator CENACE, is equipped with a functional WAMS. This system collects high-resolution synchrophasor data from PMUs strategically installed at critical buses across the grid. The operational characteristics of the SNI—combined with the availability of real-time dynamic data—create a unique environment that is both representative of challenges in other hydro-dominated systems and suitable for evaluating the robustness and effectiveness of the proposed A-VMD-based oscillation mode identification approach.

The methodology employed in this study aims to thoroughly analyze the characteristics of low-frequency oscillations in the data obtained from the WAMS system of the power grid, with a specific application to the Ecuadorian electricity system. The data was collected using the WAMS system of the independent system operator of CENACE to which all companies and members of the Ecuadorian electrical system have access through a password [31].

Power system operations typically exhibit low-frequency oscillations, which are classified into two modes: local oscillation modes and inter-area oscillation modes [32,33]:

• Local Oscillation Modes: These occur when certain synchronous machines within a defined area oscillate against one another, typically within a frequency range of 1 to 2 Hz. The characteristics of these oscillations can be observed through local measurements. In practical applications, relatively simple yet effective control measures can be implemented to dampen these oscillations. A common control strategy is the use of a conventional Power System Stabilizer (PSS), which provides a supplementary control signal to the excitation systems of generators.
• Inter-area Oscillation Modes: These oscillations involve a group of synchronous machines in one section of the system oscillating against another group in a different section, interconnected through transmission links that are prone to congestion. The typical frequency range for these modes is approximately 0.1 to 1 Hz. Inter-area oscillations occur at lower frequencies due to the higher combined inertia of coherent machines and the increased impedance of the connections between these groups.

The characteristics of inter-area oscillations are more complex and significantly different from those of local modes. The ability to effectively dampen these oscillations is limited as they are less observable and controllable. Since inter-area oscillations stem from interactions between large groups of generators, they represent a global challenge that can have widespread effects on the power system. The lack of a comprehensive system-wide perspective makes it difficult for operators to take effective measures to mitigate local oscillations and ensure adequate damping of inter-area oscillations.

3.1. Data Preprocessing

Real-time monitoring of the electric power system operation is made possible by WAMS, which stores high-dimensional data from multiple power grid buses in a database. Since measurements are recorded as they occur, they are highly susceptible to system-related fluctuations, especially when these fluctuations are significant. Consequently, the data may include values that deviate from the typical behavior of the dataset. Moreover, variations in the modal identification results can sometimes arise from the optimization procedure used by the modal identification algorithm. These deviations are outliers, or in more extreme cases, NaN values, indicating the need for data wrangling or data cleaning [6,34,35].

Techniques that identify outliers during the investigation are employed to preprocess the data. The behavior of the data can be characterized using descriptive statistics, including measures of central tendency and dispersion. This study handles NaN values using linear interpolation to manage outliers. This approach modifies the behavior of the data series. According to the literature, several techniques have been developed to detect and adjust these anomalous signal values, transforming the time series into one with a more favorable behavior. Specifically, the quartiles are statistically analyzed in this study.

In addition, trend identification has been applied in time series forecasting, as demonstrated in [36] for stock market price estimation. In [37], dynamic safety indices of the power system are constructed by analyzing the slope before fluctuations to extract trends from data collected by PMUs, such as voltage, angular separation between areas, and electrical charge.

After selecting the signal of interest, the data is preprocessed using descriptive statistics, NaN value treatment, and the MATLAB, version R2023b, function “rmoutliers” to identify outliers [38]. Outliers in this study are defined as data points that are 1.5 times higher than the upper quartile (75%) or lower than the lower quartile (25%).

WAMS monitors the operational state of the electric power system and collects dynamic information, which is then used as input for other commercial applications, such as the modal identification method. Every time a new sample is received (every 16.67 ms), this program identifies the primary oscillatory modes within the chosen signal (e.g., active power). However, various issues can impact the accuracy of this calculation, causing the results of modal identification to fluctuate. Furthermore, the high data sampling rate, which is typically employed to capture significant fluctuations, may also result in missing data or introduce noise due to the lack of an access algorithm that considers the intrinsic dynamic properties of oscillatory events.

To ensure greater reliability in the data provided by the WAMS system, algorithms are needed to more precisely quantify the signal’s trend. This is essential to avoid inaccuracies in trend estimation caused by sudden and artificial level changes resulting from calculation errors or normal system operation, including oscillatory events within the PES. This study uses the MATLAB “detrend” function to address this issue [38].

3.2. Adaptation of the VMD Technique

Through the determination of the adapted value of K, the number of IMFs in each band is identified. Each IMF’s energy is calculated, allowing for the filtering of IMF components that originate from noise. Thus, it is possible to estimate the value of K through an iterative cycle, as outlined in Algorithm 1.

Algorithm 1 Estimation of the number of modes, K.

1: Step 1: Apply FFT to the signal to obtain its frequency spectrum and estimate an initial value for K. 2: Step 2: Decompose the signal into IMFs and compute the energy of each IMF for frequency bands corresponding to oscillation modes (0.2–2 Hz). Compute the average energy for each band. 3: Step 3: Select IMFs whose energy exceeds three times the average energy of their respective band. 4: if the original signal contains only trend and noise components then 5:       Discard low-energy IMFs to remove spurious modes. 6: end if 7: Step 4: Determine the value of K as the number of selected IMFs in each band. 8: Step 5: If an IMF frequency is near a band boundary, return to Step 1 to reassess the dominant frequency.

In Step 2 of Algorithm 1, the energy of the IMF can be determined using

$$E_{\mathrm{imf}}\left(i\right)={\int }_0^T{\left|f_{ij}\left(t_j\right)\right|}^{2}dt,$$

(6)

where $E_{\mathrm{imf}}\left(i\right)$ is the energy of the ith IMF, and $f_{ij}\left(t_j\right)$ represents the ith signal at the jth sampling time.

The IMF selected in each band corresponds to the one with the highest energy and is known as the dominant modal component. Since there is at least one IMF in each band, it is important to determine its percentage of participation in the total energy, as given by

$${\eta }_i={\displaystyle \frac{E_{\mathrm{imf}}\left(i\right)}{{\sum }_{i=1}^n{E}_{\mathrm{imf}}\left(i\right)}}\times 100\%.$$

(7)

When the weight ${\eta }_i$ of the modal energy of the ith IMF exceeds a threshold value $ϵ$, this IMF is selected to represent the oscillation mode of the respective band, thereby establishing the new value of K.

It is now possible to determine the characteristic parameters of each IMF using the Prony algorithm [10]. Figure 1 schematically presents the modal identification applied to a signal obtained from the WAMS system.

Given that WAMS includes synchrophasor measurements at strategically selected buses, providing an appropriate criterion for dynamic observability, there exists a set of measurements that, through commercial applications such as WAProtector, enable the extraction of variables for the modal identification of oscillatory stability. These variables can thus be employed to characterize the system’s oscillatory behavior in real time.

Nevertheless, to ensure the reliability of such characterization, it is essential to verify that the results exhibit consistency and are not significantly affected by variability introduced through the modal identification algorithm, especially under the conditions of a highly dynamic system. Consequently, these results must undergo additional processing, specifically trend analysis, to mitigate inconsistencies before utilization. Through this process, it becomes feasible to obtain accurate real-time information regarding frequency and damping levels, thereby enabling the identification of critical oscillatory modes.

From the selected WAMS variables, a frequency distribution K is estimated. The VMD algorithm is then applied to extract the IMFs associated with each frequency. By analyzing the energy of these IMFs, their contribution to the total energy of the oscillatory mode can be assessed. This refined set of signals facilitates determining the frequency and damping levels present in the original signal.

It is important to highlight that each stage in this process necessitates the application of multivariate data analysis techniques, particularly those associated with data mining, to robustly characterize the oscillatory behavior of the system based on real-time information.

4. Numerical Results

To address the generalizability and robustness of the proposed approach, this section includes two case studies beyond the initial real-world event: (i) a synthetic oscillatory system under ideal and noisy conditions, and (ii) a second real oscillation event from Ecuador’s power system. These complementary scenarios are intended to assess the effectiveness of the A-VMD approach across a wider range of disturbances, noise levels, and dynamic characteristics.

(1) Synthetic Oscillatory System. A synthetic multi-mode signal was constructed as defined in (8), comprising three distinct sinusoidal components. A sampling interval of 0.01667 s was used over a time window of 10 s. This controlled environment allows a precise evaluation of A-VMD’s accuracy and noise resilience compared to baseline methods.  

$$\left\{\begin{array}{c}{S}_1\left(t\right)=1.0·sin(2\pi t·1.27)\hfill \\ S_2\left(t\right)=0.6·sin(2\pi t·2.7)\hfill \\ S_3\left(t\right)=0.5·sin(2\pi t·0.75)\hfill \end{array}\right.$$

(8)

Table 2. Estimated oscillation parameters for the synthetic system without noise.

Technique	Parameter	Mode 1	Mode 2	Mode 3
Synthetic Signal	Frequency (Hz)	1.27	2.7	0.75
	Amplitude	1	0.6	0.5
Matrix Pencil	Frequency (Hz)	1.26	2.65	0.77
	Error (%)	−0.79	−1.85	2.67
	Amplitude	1.011	0.617	0.509
	Error (%)	1.1	2.83	1.8
A-VMD	Frequency (Hz)	1.27	2.71	0.75
	Error (%)	0.0	0.37	0
	Amplitude	1.0	0.595	0.498
	Error (%)	0.0	−0.83	−0.4
VMD-Prony	Frequency (Hz)	1.24	2.6	0.74
	Error (%)	−2.36	−3.7	−1.33
	Amplitude	1.015	0.62	0.515
	Error (%)	1.5	3.33	3

and

Table 3. Estimated oscillation parameters for the synthetic system with SNR.

Technique	Parameter	Mode 1	Mode 2	Mode 3
Synthetic Signal	Frequency (Hz)	1.27	2.7	0.75
	Amplitude	1.1	0.66	0.55
Matrix Pencil	Frequency (Hz)	1.25	2.6	0.74
	Error (%)	−1.57	−3.7	−1.33
	Amplitude	1.11	0.671	0.559
	Error (%)	0.91	1.67	1.64
A-VMD	Frequency (Hz)	1.272	2.75	0.754
	Error (%)	0.16	1.85	0.53
	Amplitude	1.09	0.657	0.548
	Error (%)	−0.91	−0.45	−0.36
VMD-Prony	Frequency (Hz)	1.29	2.8	0.77
	Error (%)	0.78	3.7	2.67
	Amplitude	1.3	0.68	0.58
	Error (%)	1.57	3.03	5.45

summarize the estimated modal parameters for the synthetic system under noise-free and noisy conditions (SNR = 10 dB), respectively. In both scenarios, the proposed A-VMD method achieves high accuracy in estimating modal frequencies and amplitudes, outperforming traditional techniques such as Matrix Pencil and VMD-Prony in terms of relative error. Notably, the addition of white noise does not significantly degrade A-VMD’s performance, demonstrating its robustness to measurement noise.

To further assess the contribution of adaptive enhancements, an ablation study was conducted removing the IMF energy criterion and the quartile-based outlier removal step. When applying only the core VMD and Prony analyses, the estimation accuracy declined, confirming that both the adaptive mode selection and the preprocessing stage play a crucial role in improving the precision and stability of the proposed method.

It is important to note that the ’Frequency’ values reported in Table 2 and Table 3 represent the estimated modal frequencies in hertz (Hz), not the deviation from a reference value.

The selection of hyperparameters in the A-VMD framework, including the energy-based IMF selection threshold and the statistical criterion for outlier removal, was informed by empirical observations and validated through ablation analysis. In particular, the threshold of retaining IMFs whose energy exceeds three times the average within their frequency band was derived from typical energy distributions encountered in ambient and transient synchrophasor data. As evidenced in Table 2 and Table 3, removing this step notably degraded the accuracy of frequency and damping estimates, especially under noisy conditions, highlighting its role in suppressing spurious components and preserving physically meaningful modes. Similarly, outlier detection using the standard 1.5× interquartile range rule—implemented via MATLAB’s rmoutliers function—was found to enhance preprocessing stability. Disabling this step in the ablation study led to increased variability and the emergence of anomalous estimates, confirming its importance for robust modal characterization.

(2) Second Real Oscillation Event. To further support the practical validity of A-VMD, a second event recorded in Ecuador’s power grid is analyzed. The methodology was applied to a significant event in November within Ecuador’s National Interconnected System, during which several circuits associated with the international power interconnection with Colombia were disconnected. This event triggered complex dynamic behavior in the power system, including multiple oscillation modes. Among these, the most critical were the low-frequency modes, as they represent a higher risk to system stability due to their potential to propagate across vast areas and persist over time if not adequately damped.

Figure 2 presents the spectral distribution of identified oscillation modes within the WAMS database, as processed by the WAProtector application. This scatter plot visualizes the detected modes based on their frequency (Hz) on the abscissa and corresponding amplitude (MW) on the ordinate. The clustering of data points reveals distinct groupings of oscillatory behavior, indicative of prevalent system dynamics. A prominent cluster is observed, centered around a frequency of approximately 1.13 Hz, exhibiting a significant range of amplitudes, with a maximum recorded amplitude of 1.252 MW. The analysis from the WAProtector application also yielded an average damping ratio of 3.36% for these modes. Consequently, this figure illustrates the dominant frequency components and their relative magnitudes within the identified oscillation modes, suggesting the presence of significant inter-area and/or local oscillatory phenomena captured by the WAMS infrastructure.

Figure 3 presents a temporal analysis of the active power output (MW) of the Daule–Peripa power plant over a 350-s period, illustrating both ambient fluctuations and a distinct oscillatory event. The main plot shows the time series of active power, revealing a relatively stable baseline with superimposed low-amplitude variations, punctuated by a significant oscillatory transient occurring around the 120–140 s interval. To further investigate the characteristics of these distinct signal components, inset plots provide zoomed-in views. The inset labeled ‘Environment’ focuses on the period between 80 and 100 s, highlighting the inherent, low-magnitude fluctuations in active power, with amplitude variations typically less than 1 MW (revising the original 10 MW for better accuracy based on the y-axis scale). In contrast, the inset labeled ‘Oscillation’ zooms into the 120–140 s window, clearly depicting a high-amplitude oscillatory signal triggered by a disturbance. In particular, the 120–140 s interval was selected for transient analysis based on the identification of a pronounced and sustained deviation in active power measurements, clearly indicating the onset and subsequent decay of a system disturbance. This event corresponds to the disconnection of multiple international interconnection circuits, as confirmed by operational records. The amplitude of the signal during this period exceeded ±0.5 MW, which was established as an empirical threshold based on prior statistical characterization of ambient conditions. This threshold effectively distinguishes transient events from normal ambient fluctuations and helps isolate the dominant oscillatory modes, thereby enhancing the reliability and accuracy of the A-VMD-based modal identification. Thus, applying the A-VMD technique allows the extraction of modal parameters from both the ambient and oscillatory segments, enabling a quantitative comparison of their frequency, damping, and amplitude characteristics. Subsequent results will show this. The visual contrast between the ‘Environment’ and ‘Oscillation’ insets underscores the significant difference in the nature and magnitude of power swings under normal operating conditions versus during a transient oscillatory event.

Table 4. Estimated Oscillation Parameters for the SNI-Oscillatory Signal.

Technique	Parameter	Mode 1	Mode 2	Mode 3
WAMS	Frequency (Hz)	1.13	1.54	0.47
	Damping (%)	3.2	6.4	1.78
PowerFactory	Frequency (Hz)	1.132	1.52	-
	Error (%)	0.18	−1.30	-
	Damping (%)	3.25	6.5	-
	Error (%)	1.56	1.56	-
Matrix pencil	Frequency (Hz)	1.14	1.58	0.49
	Error (%)	0.88	2.6	4.26
	Damping (%)	3.9	6.8	1.36
	Error (%)	21.88	6.25	−23.60
A-VMD	Frequency (Hz)	1.134	1.531	0.46
	Error (%)	0.35	−0.58	−2.13
	Damping (%)	3.19	6.39	1.80
	Error (%)	−0.31	−0.16	1.12

presents a comparative analysis of estimated oscillation parameters for the identified oscillatory component of the SNI signal. The table details the frequency (Hz) and damping ratio (%) of the first three dominant oscillation modes as identified by four distinct techniques: WAMS, PowerFactory (a commercial power system simulation software [39]), Matrix Pencil (a signal processing algorithm), and the proposed A-VMD methodology. To quantitatively assess the accuracy of the latter three techniques, the table also includes the percentage error in both frequency and damping estimates concerning the WAMS-derived values, which serve as the benchmark in this comparison.

The results indicate that all four methods successfully identify the presence of at least two dominant oscillation modes (Mode 1 and Mode 2).

• For Mode 1, the frequencies estimated by PowerFactory (1.132 Hz) and A-VMD (1.134 Hz) exhibit very low errors (0.18% and 0.35%, respectively) relative to the WAMS-derived frequency of 1.13 Hz. The matrix pencil method shows a slightly higher frequency error of 0.88%. In terms of damping for Mode 1, PowerFactory (3.25%) and A-VMD (3.19%) again show good agreement with the WAMS value (3.2%), with errors of 1.56% and −0.31%, respectively. The Matrix Pencil method, however, presents a significantly larger damping error of 21.88%.
• For Mode 2, while all techniques identify a frequency around 1.52–1.58 Hz, the error margins vary. PowerFactory shows a small negative frequency error (−1.30%), while the matrix pencil and A-VMD exhibit positive errors (2.6% and −0.58%, respectively). Regarding damping for Mode 2 (WAMS: 6.4%), PowerFactory (6.5%) and the matrix pencil (6.8%) show relatively low errors (1.56% and 6.25%), whereas A-VMD (6.39%) demonstrates a minimal error (−0.16%).
• Mode 3 presents more varied detection. WAMS identifies a mode at 0.47 Hz with a damping of 1.78%, while PowerFactory does not detect this mode. The matrix pencil and A-VMD both identify a mode in this lower frequency range (0.49 Hz and 0.46 Hz, respectively). However, they exhibit larger frequency errors (4.26% and −2.13%) and substantial damping errors (−23.60% and 1.12%) when compared to the WAMS result. The negative damping error for the matrix pencil suggests a significant underestimation of damping for this mode.

Overall, the A-VMD technique demonstrates a strong performance in estimating the frequency and damping of the dominant Mode 1 and Mode 2, exhibiting the lowest overall error for these critical modes compared to PowerFactory and the matrix pencil. While all techniques show some discrepancies, particularly for the lower frequency Mode 3, the A-VMD method offers a more accurate and consistent modal identification for the primary oscillatory components of the SNI signal, thus highlighting the advantages of the proposed methodology for characterizing system dynamics.

Table 5. Estimated Oscillation Parameters for the SNI-Ambient Signal.

Technique	Parameters	Mode 1	Mode 2	Mode 3
WAMS	Frequency (Hz)	1.13	1.54	0.47
	Damping (%)	3.20	6.40	1.78
A-VMD	Frequency (Hz)	1.14	1.53	0.46
	Error (%)	0.88	−0.65	−2.13
	Damping (%)	3.39	6.45	1.79
	Error (%)	5.94	0.78	0.56

presents the estimated oscillation parameters (frequency and damping) for the ambient component of the SNI signal, as identified by the A-VMD technique and compared against the modal analysis capabilities of the WAProtector application (representing the WAMS-based analysis). In contrast to the findings for the oscillatory component detailed in Table 1, commercial software such as PowerFactory and the matrix pencil algorithm were unable to reliably identify distinct oscillation modes within the lower-energy ambient signal. This highlights a potential limitation of traditional modal analysis techniques when applied to signals characterized by subtle dynamic variations.

The A-VMD technique, however, successfully detected three oscillation modes in the ambient signal. Comparing these results with the modes identified by WAMS for the oscillatory component (Table 1) reveals interesting insights. Mode 1 identified in the ambient signal by A-VMD (1.14 Hz, 3.39% damping) closely corresponds in frequency to the dominant Mode 1 observed during oscillatory events (1.13 Hz by WAMS), suggesting a persistent natural frequency within the system that is excited even under normal operating conditions, albeit with different energy levels and potentially slightly altered damping characteristics. The damping for this mode identified by A-VMD (3.39%) is also comparable to the WAMS result for the oscillatory Mode 1 (3.2%).

Similarly, Mode 2 identified by A-VMD in the ambient signal (1.53 Hz, 6.45% damping) shows a close frequency proximity to the Mode 2 observed by WAMS during oscillations (1.54 Hz), further supporting the existence of inherent system-wide oscillation frequencies. The damping values for this mode are also remarkably consistent between the ambient (6.45%) and oscillatory (6.4%) scenarios.

Mode 3 detected by A-VMD in the ambient signal (0.46 Hz, 1.79% damping) also aligns well with the frequency of Mode 3 identified by WAMS during oscillations (0.47 Hz), reinforcing the notion of these being intrinsic system modes. The damping values for this lower-frequency mode are also very similar (1.79% by A-VMD vs. 1.78% by WAMS).

The successful identification of these consistent modal frequencies in the ambient signal by the A-VMD technique, where other methods failed, underscores its enhanced sensitivity and ability to extract valuable information about the underlying system dynamics even from low-energy fluctuations. This capability allows a more comprehensive understanding of the power system’s natural oscillation modes, which are present not only during significant disturbances but also as subtle variations in the normal operating state. The close agreement in the identified frequencies across both ambient and oscillatory conditions suggests that these are fundamental characteristics of the SNI, whose excitation level and potentially damping are modulated by system events.

Finally, Figure 4 presents the IMFs of the oscillatory signal obtained through the application of the A-VMD technique, focusing on the time window of the oscillatory event (119–125 s). The top panel displays the original oscillatory signal segment. The subsequent panels show the first three IMFs (IMF1, IMF2, and IMF3) extracted by the A-VMD, which represent the constituent oscillatory components of the signal, ordered by their central frequency content (with IMF1 typically containing the highest frequency components).

By decomposing the oscillatory signal into these individual IMFs, the A-VMD allows a detailed examination of the temporal evolution and energy contribution of each identified mode. While Table 1 provided the estimated frequency and damping parameters for these modes, Figure 4 offers a time-domain visualization of their activity during the transient event. For instance, the prominent oscillations observed in the original signal around 122 s are decomposed into contributions across the different IMFs.

Relating this to Table 1, IMF1 likely corresponds to the higher frequency modes (Mode 2 around 1.53–1.54 Hz) identified by A-VMD and WAMS, as it exhibits relatively faster oscillations. IMF2 and IMF3, with their lower frequency content, likely represent the contributions of the dominant Mode 1 (around 1.13–1.14 Hz) and the lower-frequency Mode 3 (around 0.46–0.47 Hz), respectively. The amplitudes of these IMFs over time provide insights into the energy associated with each mode during the oscillation. Periods of higher amplitude in a specific IMF indicate a greater energy contribution of that particular mode to the overall signal at that time.

The ability of A-VMD to decompose the signal into these physically meaningful IMFs allows a more nuanced understanding of the oscillatory behavior compared to traditional techniques that might only provide aggregated modal parameters. The temporal separation of the modes, as visualized in Figure 4, can be further analyzed to understand the triggering and decay characteristics of each oscillatory component, potentially revealing the presence of modes with higher damping that might not be easily discernible in the raw signal or through less sophisticated modal analysis methods. This decomposition complements the frequency and damping information presented in Table 1 by providing a time-resolved perspective on the energy dynamics of the identified oscillation modes.

5. Discussion

Maintaining small-signal stability is paramount for the continuous and reliable operation of interconnected power systems (PSs), and modal analysis has traditionally served as a cornerstone for this purpose. Critical modes can be identified and their residues quantified by extracting the frequency, damping ratios (derived from eigenvalues), and mode shapes of system oscillations. This information is crucial for strategically locating and tuning power system stabilizers (PSSs) to damp these oscillations effectively, a central theme explored in this thesis.

The advent of synchronized phasor measurement technology, implemented through PMUs and integrated within WAMS, provides an unprecedented opportunity for real-time monitoring of power system dynamics. Leveraging WAMS data for enhanced small-signal stability assessment and control is a significant and active research domain [4,40,41], particularly focusing on utilizing synchrophasor measurements from generators with high observability and controllability for PSS optimization.

The increasing interconnection of power systems, driven by economic and reliability considerations, introduces complexities in managing small-signal oscillations. Variations in operating points can significantly impact the damping of these inter-area and local modes, making their robust control a critical research challenge. Consequently, the development of advanced methodologies that effectively utilize synchrophasor signals for oscillation control is essential. The A-VMD technique proposed in this work represents a significant contribution in this direction, demonstrating enhanced capabilities in identifying oscillation modes, particularly in scenarios where traditional methods struggle (as evidenced by the ambient signal analysis in Table 5).

The application of the proposed A-VMD methodology, as illustrated in Figure 4 through the decomposition of the oscillatory signal into its IMFs, offers a time-resolved perspective on the energy contribution of different oscillatory components. This complements the modal parameter estimations presented in Table 1, where A-VMD demonstrated comparable or superior accuracy in identifying the frequency and damping of dominant modes (Mode 1 and Mode 2) compared to WAMS data analysis, PowerFactory, and the matrix pencil. Notably, A-VMD’s ability to extract modes from the low-energy ambient signal (Table 5), where other techniques failed, underscores its sensitivity to subtle system dynamics and its potential for continuous, proactive monitoring. The consistency in the frequencies of the modes identified by A-VMD across both oscillatory and ambient conditions suggests the accurate capture of inherent system-wide oscillation frequencies.

While the present study focuses on a hydropower-dominated system, the proposed A-VMD methodology is equally applicable to power systems with high penetration of inverter-based renewable energy sources (IBRESs), such as solar photovoltaic (PV) energy. These systems present unique challenges for oscillation mode analysis due to their reduced mechanical inertia, their rapidly varying generation profiles, and the predominance of powered electronic interfaces. Such conditions often result in the emergence of low-damping oscillatory modes that are difficult to capture with conventional model-based techniques, which rely on accurate dynamic models and assumptions that may not hold in highly variable scenarios. The A-VMD approach, by contrast, operates directly on synchrophasor data and incorporates an adaptive, energy-driven decomposition framework, which enhances its sensitivity to weak and masked modes under both ambient and transient conditions. This data-driven adaptability, combined with robustness to noise and non-stationary behaviors, positions A-VMD as a promising tool for modal analysis and small-signal stability monitoring in future grids with high levels of solar PV energy and other IBRESs.

While the proposed methods necessitate comprehensive system analysis and real-time data processing for continuous monitoring, the potential for integrating model reduction techniques remains a crucial area for future research. Reducing the computational burden associated with advanced signal processing techniques such as A-VMD and the subsequent data mining for control applications would enhance the scalability and real-time applicability of these innovative methodologies in large-scale power systems.

6. Conclusions

In this paper, we addressed the critical need for robust small-signal stability assessment in modern power systems by proposing and implementing an enhanced modal analysis framework leveraging the A-VMD technique on WAMS data. We demonstrated the effectiveness of A-VMD in accurately identifying dominant oscillation modes from the oscillatory signal of the SNI system, achieving comparable or superior accuracy in frequency and damping estimation relative to traditional methods and commercial software. Notably, our application of A-VMD to the ambient system conditions revealed its unique ability to extract inherent system oscillation modes undetectable by conventional techniques, highlighting its enhanced sensitivity to subtle system dynamics. The time-domain decomposition of the oscillatory signal into IMFs provided further insights into the temporal evolution and energy distribution of these identified modes.

The promising results obtained by applying A-VMD underscore its potential as a powerful tool for monitoring and analyzing advanced power systems. Our proposed methodology offers a more nuanced understanding of system dynamics by effectively characterizing both significant oscillatory events and the underlying modal behavior present during regular operation. This capability is particularly relevant for adapting to interconnected power grids’ evolving and increasingly complex nature. The ability to consistently identify key system frequencies across different operating conditions suggests that A-VMD can contribute to more accurate and proactive stability assessments, paving the way for improved control strategies.

Building upon the findings of this research, future endeavors could focus on the real-time implementation of the proposed A-VMD-based modal analysis framework for continuous online monitoring. Investigating the integration of these identified modal parameters with adaptive control schemes, such as PSS tuning, presents a promising avenue. Furthermore, exploring the scalability and performance of the methodology in larger, more complex power systems and under high penetration of renewable energy sources would be a good direction for further research. Finally, the application of machine learning techniques for automated mode tracking, classification, and prediction based on the A-VMD-decomposed signals could significantly enhance the proactive management of power system stability.