Higher-Order Dynamic Mode Decomposition to Identify Harmonics in Power Systems

Abstract

The proliferation of renewable energy sources and distributed generation systems interfaced to the grid by power electronics systems is forcing us to better understand the issues arising due to the quality of electrical signals generated through these devices. Understanding and monitoring these harmonics is crucial to ensure the smooth and seamless operation of these networks, as well as to protect and manage the renewable energy sources-based power system. In this paper, we propose an advanced method of dynamic modal decomposition, called Higher-Order Dynamic Mode Decomposition (HODMD), one of the recently proposed data-driven methods used to estimate the frequency/amplitude and phase with high resolution, to identify the harmonic spectrum in power systems dominated by renewable energy generation. In the proposed method, several time-shifted copies of the measured signals are integrated to create the initial data matrices. A hard thresholding technique based on singular value decomposition is applied to eliminate ambiguities in the measured signal. The proposed method is validated and compared to Synchrosqueezing Transform based on Short-Time Fourier Transform (SST-STFT) and the Concentration of Frequency and Time via Short-Time Fourier Transform (ConceFT-STFT) using synthetic signals and real measurements, demonstrating its practical effectiveness in identifying harmonics in emerging power networks. Finally, the effectiveness of the proposed methodology is analyzed on the energy storage-based laboratory-scale microgrid setup using an Opal-RT-based real-time simulator.

1. Introduction

The growing integration of renewable energy production systems into power grids presents major challenges, not only due to the intermittent nature of renewable sources but also because of the power electronic interfaces that connect them to the grid [1]. Understanding the interactions between these converters and the grid is crucial for assessing signal quality, ensuring system stability, and preventing adverse phenomena in converter-dominated networks.

With the increasing penetration of power electronic converters, electromagnetic transient (EMT) studies have become indispensable. These studies capture fast, short-duration phenomena and require high-resolution time–frequency analysis tools for accurate characterization. EMT simulations are widely recommended by international standards such as NERC Project 2022-04 to represent system behavior across a wide frequency range. However, their interpretation is often limited by the resolution constraints of conventional spectral analysis techniques [2].

Several methods have been proposed for power system signal analysis, including Fourier techniques [3,4], optimization-based methods, adaptive filters, zero-crossing techniques [5], and neural network approaches [6]. Despite their usefulness, these methods exhibit limitations such as poor robustness to noise, dependency on prior frequency knowledge, and difficulty in detecting interharmonics or subsynchronous components. They also struggle to accurately estimate weakly damped and time-varying components. Importantly, phase estimation remains a critical challenge: Unreliable phase tracking can lead to erroneous signal reconstructions and misinterpretation of system dynamics.

To address these issues, Prony’s method and its variants have been used to estimate modal frequency, damping, amplitude, and phase from PMU ringdown data. Although effective, Prony analysis remains sensitive to noise, requires sufficiently excited modes, and incurs significant computational cost when exploring multiple model orders [7]. These limitations have motivated the exploration of data-driven approaches such as Dynamic Mode Decomposition (DMD), which derives models directly from time-series snapshots [8]. Originally developed in fluid dynamics, DMD has since been applied in oscillation analysis [9,10], inertia estimation [11], and state estimation frameworks using Kalman filtering [12].

Recent studies have demonstrated DMD’s effectiveness in estimating frequency and amplitude in electrical systems [13,14]. For instance, ref. [15] combined Variational Mode Decomposition (VMD) with DMD to enhance robustness under noisy conditions, using VMD for decomposition and DMD for sparse feature extraction. However, the method is computationally intensive, sensitive to the number of modes, and cannot directly isolate DC components, while VMD may fail under highly non-stationary conditions with abrupt onsets [16].

Due to their computational simplicity, Fourier-based methods are frequently employed [17]; however, spectrum leakage and boundary effects can reduce accuracy, especially in non-stationary or transient environments. Higher-Order Dynamic Mode Decomposition (HODMD) addresses these shortcomings by augmenting the snapshot matrix with multiple time shifts, increasing the effective state dimension and improving spectral resolution. HODMD has been shown to outperform standard DMD by detecting weak or closely spaced modes and filtering out low-energy dynamics [18], making it well suited for the analysis of quasi-periodic and transient signals in converter-dominated systems [19].

In this paper, we apply HODMD to analyze a wide range of signals from both conventional and converter-dominated power systems. We compare its performance against two advanced time–frequency analysis techniques: Synchrosqueezing Transform-based Short-Time Fourier Transform (SST-STFT) [20,21] and the Concentration of Frequency and Time via SST (ConceFT-STFT) [22]. Both stationary and non-stationary synthetic signals are used to benchmark accuracy. Experimental validation is performed using a detailed battery energy storage system model with switching effects and a real-time OPAL-RT OP4510/HYPERSIM platform. Finally, we demonstrate the method’s capability on real-world PMU data from the Hydro-Québec grid.

The main contributions of this paper are summarized as follows:

• Introduction of a robust harmonic spectrum analysis framework based on HODMD, maintaining accuracy under stressed operating conditions such as transients, microgrids, and distorted waveforms;
• Extension of harmonic analysis to non-stationary signals, enabling accurate extraction of frequency, amplitude, and phase for both fundamental and harmonic components;
• Development of a sliding window HODMD implementation for high resolution time–frequency tracking with amplitude;
• Validation of the proposed method through simulation and real-time experiments, demonstrating robustness to switching harmonics and noise in both simulated and real-world grid operation and monitoring environments.

2. Dynamic Mode Decomposition (DMD)

DMD is a powerful data-driven technique for analyzing both stationary and non-stationary signals arising from complex dynamical systems. In the context of power systems, such signals often consist of multiple components, harmonic distortions, interharmonic oscillations, and DC offsets, each contributing to the overall system behavior in a distinct way. These composite signals can be mathematically expressed as follows:

$$v_k=\sum _{j=1}^N{\varphi }_j{a}_j{e}^{{\lambda }_{j}k},$$

(1)

where $v_k$ is a time-dependent vector of size J, ${\varphi }_j$ are normalized spatial modes, and $a_j$ are mode amplitudes. The term ${\lambda }_j$ represents the damping ratio or growth rate. This formulation elegantly captures a wide spectrum of signal features typically seen in power systems, such as the following:

• Harmonics: integer multiples of a fundamental frequency due to nonlinear loads;
• Interharmonics: frequencies that lie between integer harmonics, often resulting from variable speed drives or power electronic switching;
• DC Components: steady-state offsets that may arise from asymmetrical faults or rectified signals.

By decomposing these signals into mode-specific temporal and spatial dynamics, DMD provides a framework that not only enhances interpretability but also enables targeted diagnostics, filtering, and system identification, particularly under transient or non-periodic conditions.

The general waveform of interest for this study is a single-phase sinusoidal signal (expressed in per unit), with superimposed DC, interharmonic, and fundamental harmonic components

$$\begin{array}{cc}\hfill x\left(k\right)=& x_{DC}+x_p\cos \left(2\pi f_{c}kT+{\theta }_p\left(k\right)\right)\hfill \\ \hfill & +x_{\mathrm{ih}}\cos \left(2\pi f_{\mathrm{ih}}kT+{\theta }_{\mathrm{ih}}\left(k\right)\right)\hfill \\ \hfill & +\sum _{m=2}^M{x}_m\cos \left(2\pi m{f}_{c}kT+{\theta }_m\left(k\right)\right)\hfill \end{array}$$

(2)

where

• $x_{DC}$: DC value with/without decaying;
• $x_m$, ${\theta }_m$: magnitude and phase of the $m\mathrm{th}$ harmonic;
• $x_p$, $f_c$, ${\theta }_p$: magnitude, frequency, and angle of the fundamental frequency in Hz;
• $x_{\mathrm{ih}}$, $f_{\mathrm{ih}}$, ${\theta }_{\mathrm{ih}}$: magnitude, frequency, and angle of the inter-harmonic component.

For simplicity, but without loss of generality, we assume the signal contains a single interharmonic. This formalism can be easily extended to multiple interharmonics both above and below the fundamental frequency.

2.1. Standard DMD Assumption

The classical DMD approach operates under the premise that a linear operator R governs the progression of flow snapshots $v_k$ (state vectors) to their successive snapshots $v_{k+1}$, a concept commonly referred to as the Koopman assumption, which can be expressed as follows:

$$\begin{array}{cc}\hfill V_{k+1}& =R{V}_k\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill V_k& =[V_1,V_2,V_3,\dots ,V_k]\hfill \end{array}$$

$$\begin{array}{cc}\hfill V_{k+1}& =[V_2,V_3,V_4,\dots ,V_{k+1}]\hfill \end{array}$$

(4)

where R is the Koopman matrix, considered independent of k. The general solution is approximated by

$$v_k\approx \sum _{j=1}^M{\varphi }_j{b}_j{\lambda }_j^k,$$

(5)

where ${\varphi }_j$ and ${\lambda }_j$ are eigenvectors and eigenvalues of R, respectively.

DMD is capable of isolating static features from dynamic behaviors; however, its formulation is inherently first order, limiting its ability to capture higher-order temporal correlations and resolve closely spaced spectral components. To address this limitation, Higher-Order Dynamic Mode Decomposition (HODMD) was developed by augmenting the snapshot matrix with multiple time shifts, thereby increasing the effective state dimension. This enhancement improves spectral resolution, enables the detection of weak and closely spaced modes, and enhances robustness to noise, making HODMD particularly well suited for the analysis of non-stationary signals and systems with multiple time scales [23].

2.2. Koopman Operator and Singular Value Decomposition (SVD)

The Koopman Operator underlies DMD and was first introduced in 1931 by Koopman [24]. Its defining characteristic is the focus on how functions of the states, called observables, evolve over discrete time:

$$g\left(x_{k+1}\right)=\mathcal{K}g\left(x_k\right),$$

(6)

where $\mathcal{K}$ is the Koopman Operator. SVD is a crucial matrix decomposition technique employed in HODMD for data reduction. Given a snapshot matrix X, SVD expresses it as

$$X=US{V}^T,$$

(7)

where U and V are the left and right singular vectors, respectively, and S contains the singular values.

3. Higher-Order Dynamic Mode Decomposition (HODMD)

HODMD extends DMD to cases where the spatial complexity M is smaller than the spectral complexity N. This technique merges the standard DMD with Takens’ delayed embedding theorem. Each snapshot is assumed to depend linearly on the most recent snapshot and on $d-1$ additional past snapshots.

We consider the following snapshot expression:

$$v\left(t\right)\approx \sum _{n=1}^N{a}_n{u}_n{e}^{({\delta }_n+i{\omega }_n)(t-t_1)},t_1\le t\le t_1+T$$

(8)

where $a_n$ is the magnitude of the eigenvector $u_n$, and $e^{({\delta }_n+i{\omega }_n)(t-t_1)}$ is the eigenvalue of the matrix A.

The HODMD method, also referred to as DMD-d, is based on the high-order Koopman assumption, where the flow snapshot $v_{k+d}$ can be expressed as a function of the preceding “d” flow snapshots, as follows:

$$\begin{array}{cc}\hfill v_{k+d}& \simeq R_1{v}_k+R_2{v}_{k+1}+\cdots +R_d{v}_{k+d-1},\hfill \\ \hfill & \mathrm{for}k=1,\dots ,K-d.\hfill \end{array}$$

(9)

where d is a tunable constant and $R_k$ represents the Koopman operators, which are linear and encapsulate the system dynamics. This formulation constitutes the core of the HODMD approach, known for delivering highly accurate results even when applied to complex datasets. When $d=1$, the high-order Koopman assumption simplifies to the standard DMD algorithm. Further methodological details can be found in [23,25].

Algorithm of HODMD

HODMD computes the frequencies, growth rates, and DMD modes in four main steps, as illustrated in Figure 1. To facilitate reproducibility and parameter tuning, Table 1 summarizes practical guidelines for selecting the key parameters.

Figure 1. Flow diagram of HODMD algorithm.

Table 1. Practical guidelines for tuning HODMD parameters.

Parameter	Practical Range	Guideline for Selection
d (Delay Embedding Dimension)	Depends on signal complexity and noise level; trial-and-error tuning	Start with $d\approx$ twice the expected number of dominant modes. Increase d until the reconstruction error converges. Stop increasing if the SVD becomes ill conditioned.
${\epsilon }_1$ (SVD Tolerance)	${10}^{-6}$–${10}^{-12}$	Choose ${\epsilon }_1$ to retain at least $95\%$ of the system energy. If runtime becomes excessive without improving accuracy, increase ${\epsilon }_1$.
${\epsilon }_{\mathrm{SVD}}$ (Mode Selection Threshold)	${10}^{-4}$–${10}^{-2}$	Select a threshold that filters out noise like modes while retaining physically meaningful weak modes.

The parameter d has the most significant impact on the accuracy of the identified modes and the quality of the reconstruction. This parameter is tuned using a trial-and-error approach. For this work, we developed an automated routine that iteratively tests several candidate values of d, performs the HODMD decomposition, and selects the value that minimizes the root mean square error between the reconstructed and original signals. This procedure ensures that the chosen d leads to an optimal trade-off between reconstruction accuracy and numerical stability.

For ${\epsilon }_1$ and ${\epsilon }_{\mathrm{SVD}}$, their effect is more related to noise rejection than to frequency accuracy. A too-loose tolerance (${\epsilon }_1$ too high) may discard physically relevant modes, while a too-strict tolerance can keep spurious noise components and increase computational cost. Similarly, ${\epsilon }_{\mathrm{SVD}}$ controls the selection of dynamically significant modes: it should be set to remove noise like modes but retain weaker harmonics of interest. Sensitivity analyses showed that once these parameters are within reasonable ranges, their effect on frequency estimates is minor compared to d.

Step 1: First Dimension reduction

We apply to the snapshot matrix defined in Equation (4) the SVD to the matrix $V_K$ based on a tolerance ${\epsilon }_1$ given by the user as follows:

$$V^K=U{\tilde{V}}_1^K\mathrm{where}{\tilde{V}}_1^K=S{V}^T,$$

(10)

where the number of retained modes N is defined as ${\sigma }_{N+1}/{\sigma }_1\le {\epsilon }_{\mathrm{SVD}}$, where ${\sigma }_1,\dots ,{\sigma }_N$ are the singular values and the threshold ${\epsilon }_{\mathrm{SVD}}$ is selected according to the level of noise in the data.

${\tilde{V}}_1^K$ is called the reduced snapshot matrix.

Step 2: Second Dimension reduction

In the second step, we apply the higher-order Koopman assumption defined in Equation (9) to the reduced snapshot matrix as follows:

$${\tilde{V}}_2^K\simeq {\tilde{R}}_1{\tilde{V}}_1^{K-d}+{\tilde{R}}_2{\tilde{V}}_2^{K-d+1}+\cdots +{\tilde{R}}_d{\tilde{V}}_d^K,$$

(11)

where ${\tilde{R}}_i=U^T{R}_{i}U$.

This equation can be represented using the reduced snapshot matrix and the modified Koopman matrix $\tilde{\mathbf{R}}$ as follows:

$${\tilde{V}}_1^{K-d+1}=\tilde{\mathbf{R}}{\tilde{V}}_2^{K-d},$$

(12)

where

$${\tilde{V}}_1^{K-d}=\left[\begin{array}{c}{\tilde{V}}_1^{K-d}\\ {\tilde{V}}_2^{K-d+1}\\ ⋮\\ {\tilde{V}}_d^K\end{array}\right],{\tilde{V}}_2^{K-d+1}=\left[\begin{array}{c}{\tilde{V}}_{d+1}^{K+1}\end{array}\right],\tilde{\mathbf{R}}=\left[\begin{array}{ccccc}0& I& 0& \dots & 0\\ 0& 0& I& \dots & 0\\ ⋮& ⋮& \ddots & \ddots & ⋮\\ 0& 0& \dots & 0& I\\ {\tilde{R}}_1& {\tilde{R}}_2& \dots & {\tilde{R}}_{d-1}& {\tilde{R}}_d\end{array}\right]$$

A second dimensionality reduction is carried out on the matrix containing the reduced snapshots using SVD and the tolerance ${\epsilon }_{\mathrm{SVD}}={\sigma }_{N^{\prime }+1}/{\sigma }_1\le {\epsilon }_{\mathrm{SVD}}$, where $N^{\prime }$ is the number of retained SVD modes and ${\sigma }_i$ are the singular values. This truncation yields

$${\tilde{V}}_1^{K-d+1}\simeq \tilde{U}\tilde{S}{\tilde{V}}_1^T,\mathrm{with}{\tilde{V}}_1^{K-d+1}=\tilde{S}{\tilde{V}}^T.$$

(13)

This step is completed through pre-multiplying Equation (12) by ${\tilde{U}}^T$, and invoking Equation (13), it gives

$${\tilde{V}}_2^{K-d+1}={\tilde{R}}^{*}{\tilde{V}}_1^{K-d},$$

(14)

such that ${\tilde{R}}^{*}\in {\mathbb{R}}^{N^{\prime }\times N^{\prime }}$ is the new Koopman matrix defined as ${\tilde{R}}^{*}={\tilde{U}}^T\tilde{R}\tilde{U}$.

Step 3: computing the DMD modes, frequencies, and growth rates.

This is performed by applying SVD on the matrix ${\tilde{V}}_1^{K-d}$:

$${\tilde{V}}_1^{K-d}=UA{V}^T.$$

(15)

Then, we substitute Equation (16) in Equation (14) and multiply the result by $V{A}^{-1}{U}^T$ to obtain

$${\tilde{R}}^{*}={\tilde{V}}_{2}V{A}^{-1}{U}^T,$$

(16)

Once the matrix ${\tilde{R}}^{*}$ has been calculated, the reduced DMD expansion for the reduced snapshots Equation (10) can be computed as follows:

$${\tilde{\mathbf{v}}}_k=\sum _{m=1}^M{\widehat{a}}_m{\tilde{u}}_m{e}^{({\delta }_m+i{\omega }_m)t_k},\mathrm{for}k=1,\dots ,K.$$

(17)

The reduced DMD modes ${\tilde{u}}_m$ were calculated by keeping the first M elements of the vector ${\widehat{q}}_m=\tilde{U}{q}_m$, where $q_m$ represents the eigenvectors of $\tilde{R}$ and the associated eigenvalues ${\mu }_m$ provides the frequencies ${\omega }_m$ and growth rates ${\delta }_m$ by the following expression:

$${\delta }_m+i{\omega }_m=\log \left({\mu }_m\right)/\Delta t.$$

(18)

The identification of the dominant modes in HODMD relies on two complementary mechanisms:

• SVD-Based Truncation: During the first step of HODMD, singular value decomposition (SVD) is applied to the snapshot matrix. This step retains only the most energetic modes, effectively filtering out low-energy noise components.
• Reconstruction Error Check: After mode selection, the reconstructed signal is compared to the original data, and the root mean square error (RMSE) is computed. A high RMSE indicates that a relevant mode may have been discarded or misclassified. In such cases, the parameter d or the tolerance ${\epsilon }_{\mathrm{SVD}}$ is adjusted until the reconstruction error is minimized, ensuring recovery of the missing dominant mode.

4. Results

4.1. Performance Evaluation on Synthetic Signal

4.1.1. Case Study 1: Stationary Signal

In this case study, the performance of HODMD, SST-STFT, and ConceFT-STFT was evaluated on a synthetic stationary signal. We consider a multi modal voltage signal with a DC offset and interharmonic components, expressed as

$$\begin{array}{cc}\hfill V\left(t\right)=& 0.5+0.1\sin \left(2\pi t·30+{\displaystyle \frac{\pi }{2}}\right)\hfill \\ \hfill & +\sin \left(2\pi t·60-{\displaystyle \frac{\pi }{3}}\right)\hfill \\ \hfill & +0.05\sin \left(2\pi t·90+0.22\right)\hfill \\ \hfill & +0.05\sin \left(2\pi t·120+{\displaystyle \frac{\pi }{4}}\right)\hfill \\ \hfill & +0.01\sin \left(2\pi t·180+\pi \right)\hfill \\ \hfill & +0.01\sin \left(2\pi t·240-0.52\right)\hfill \end{array}$$

(19)

The complex analytic form of this signal can be written as

$$\begin{array}{cc}\hfill \tilde{V}\left(t\right)=& 0.5+0.1e^{j\left(2\pi ·30t+{\displaystyle \frac{\pi }{2}}\right)}+1.0e^{j\left(2\pi ·60t-{\displaystyle \frac{\pi }{3}}\right)}+0.05e^{j\left(2\pi ·90t+0.22\right)}\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & +0.05e^{j\left(2\pi ·120t+{\displaystyle \frac{\pi }{4}}\right)}+0.01e^{j\left(2\pi ·180t+\pi \right)}+0.01e^{j\left(2\pi ·240t-0.52\right)}\hfill \end{array}$$

(21)

$$V\left(t\right)=\Im \left\{\tilde{V}\left(t\right)\right\}$$

The identified frequency components and their corresponding amplitudes using the three methods are presented in Table 2.

Table 2. Comparison of the three spectral analysis methods.

Method	freq (Hz)	Amp (pu)	$\mathit{\theta }$ (°)	RMSE freq (%)	RMSE Amp (%)	RMSE $\mathit{\theta }$ (%)
HODMD	0	0.5	N/A	$1.71\times {10}^{-13}$	$1.266\times {10}^{-11}$	N/A
	30	0.1	90	$1.42\times {10}^{-13}$	$5.953\times {10}^{-11}$	$1.258\times {10}^{-11}$
	60	1	60	$2.842\times {10}^{-13}$	$3.135\times {10}^{-11}$	$5.001\times {10}^{-11}$
	90	0.05	12.6	$1.579\times {10}^{-13}$	$2.266\times {10}^{-11}$	$2.163\times {10}^{-10}$
	120	0.05	45	$1.539\times {10}^{-13}$	$6.4\times {10}^{-11}$	$5.649\times {10}^{-11}$
	180	0.01	180	$1.578\times {10}^{-14}$	$7.265\times {10}^{-11}$	$1.173\times {10}^{-12}$
	240	0.01	−29.79	$2.013\times {10}^{-13}$	$6.7637\times {10}^{-11}$	$3.074\times {10}^{-10}$
SST-STFT	0.0914	0.5026	N/A	$9.1\times {10}^{-2}$	$6.3\times {10}^{-1}$	N/A
	30.0849	0.0911	89.99	$2.7\times {10}^{-1}$	78	$1.938\times {10}^{-5}$
	60.0718	1	60	$1.145\times {10}^{-2}$	$3.16\times {10}^{-6}$	$6.866\times {10}^{-7}$
	90.0718	0.0458	12.609	$1.788\times {10}^{-1}$	$7.6$	$5.44\times {10}^{-5}$
	120.1568	0.0489	44.99	$1.425\times {10}^{-13}$	$4.232\times {10}^{-1}$	$2.531\times {10}^{-8}$
	180.1437	0.0101	180	$8.038\times {10}^{-2}$	$1.131$	$2.517\times {10}^{-10}$
	240.1306	0.0101	−29.79	$5.566\times {10}^{-2}$	$1.114$	$2.23\times {10}^{-4}$
ConceFT-STFT	N/A	N/A	N/A	N/A	N/A	N/A
	30.0027	0.0844	−0.638	$9.0433\times {10}^{-3}$	$15.6$	100.708
	60.0053	1	9.166	$8.883\times {10}^{-3}$	$5.17\times {10}^{-6}$	115.276
	90.008	0.0427	−5.248	$8.0377\times {10}^{-3}$	$14.586$	141.69
	120.0107	0.05008	5.135	$9.225\times {10}^{-3}$	$1.601\times {10}^{-1}$	88.59
	180.01607	0.0106	−0.036	$6.444\times {10}^{-3}$	$5.879$	99.97
	240.0214	0.0107	−1.115	$8.917\times {10}^{-3}$	$6.773$	94

The percentage error between the predicted and the actual values is computed as follows:

$$\mathrm{error}(\%)={\displaystyle \frac{\left|y-y_{\mathrm{ref}}\right|}{y_{\mathrm{ref}}}}\times 100$$

(22)

This metric quantifies the relative deviation of the identified value y from the reference value $y_{\mathrm{ref}}$ in percentage form.

The normalized root mean square error (RMSE%) can be expressed as

$$\mathrm{RMSE}(\%)={\displaystyle \frac{\sqrt{{\displaystyle {\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(y_i-y_{\mathrm{ref},i}\right)}^2}}}{y_{\mathrm{ref}}^{\mathrm{norm}}}}\times 100$$

(23)

where $y_{\mathrm{ref}}^{\mathrm{norm}}$ is a chosen normalization value, typically the maximum value, RMS value, or mean of the reference signal $y_{\mathrm{ref}}$. This normalization allows expressing the RMSE as a percentage of the signal magnitude.

From Table 2, it can be observed that for the measured signal, HODMD provides the highest accuracy in estimating frequencies, amplitudes, and phases. ConceFT-STFT provides better frequency resolution than SST-STFT but less reliable amplitude and phase information. This can be a supplemental diagnostic tool for spectrum characterization. Overall, considering the phase results in particular, HODMD clearly outperforms the other methods.

The reconstruction error of HODMD, quantified using the root mean square error, is $\mathrm{RMSE}=2.71\times {10}^{-13}$, indicating an almost perfect match between the reconstructed and original signals. It is important to note that RMSE cannot be computed for the SST-STFT and ConceFT-STFT, as these techniques do not reconstruct the time-domain signal directly but instead provide an energy distribution in the time–frequency plane.

4.1.2. Case Study 2: Hauer’s Signal

In this case study, HODMD is applied to the well-known Hauer signal, shown in Figure 2, which serves as a benchmark test case for non-stationary and multi-harmonic time-series analysis in power systems. The Hauer signal consists of three primary sinusoidal components: the fundamental (H1) at $58.92 \mathrm{Hz}$ and two off-nominal harmonics, H2 and H3, with nominal frequencies of $117.84 \mathrm{Hz}$ and $176.76 \mathrm{Hz}$, respectively. The magnitudes of H2 and H3 are set to $2\%$ and $5\%$ of the fundamental amplitude, respectively. The signal is segmented into seven intervals of $2 \mathrm{s}$ each, with each interval exhibiting a distinct harmonic composition, as described in [26]. This design introduces abrupt spectral transitions, making the Hauer signal an ideal candidate for evaluating advanced time–frequency analysis techniques.

Figure 2. Phase a of Hauer signal [26].

The HODMD results (see Figure 3) demonstrate precise tracking of the fundamental and harmonic components across the various time intervals. For each analysis window, the frequency, amplitude, and phase evolution of H1, H2, and H3 are individually extracted and analyzed. During intervals where only the fundamental is present (0–8 s), the algorithm consistently identifies a dominant mode at $58.92 \mathrm{Hz}$ with a stable amplitude close to unity. In subsequent intervals, the appearance or disappearance of harmonics (H2 or H3) is correctly detected, and the corresponding frequency and amplitude estimates closely match the expected theoretical values.

Figure 3. Result of HODMD on Hauer signal (part 1).

Notably, the phase analysis of each harmonic reveals a quasi-sinusoidal evolution corresponding to the initialization and phase transitions described in the initial test case. The ability of HODMD to clearly separate and track the dynamics of each harmonic, in both the amplitude and phase, even in the presence of abrupt spectral changes, highlights its robustness and effectiveness relative to conventional time frequency analysis techniques.

Overall, the results obtained from the Hauer signal confirm the capability of HODMD to resolve complex time-varying spectral structures with high precision. Leveraging Hauer’s signal, which is widely recommended for benchmarking phasor harmonic estimation under adverse conditions that typically challenge Fourier-based approaches, the proposed algorithm demonstrates strong competitiveness with the benchmark method presented in [26]. Importantly, HODMD offers advantages in terms of implementation simplicity and interpretability, making it a compelling alternative for robust harmonic analysis.

4.1.3. Case Study 3: Non-Stationary Signal

In this case study, we modulated a non-stationary synthetic signal containing the same components as in Case Study 1. However, in this scenario, the frequency components vary with time, as summarized in Table 3. The simulation was run for $1 \mathrm{s}$ with a sampling frequency of $1024 \mathrm{Hz}$.

Table 3. Parameter variation.

Frequency	Parameter Variation	ROCOF	Initial Value	Final Value
f1	$60+t$	+1	60	61
f2	$30+2t$	+2	30	32
f3	$120-2t$	−2	120	118
f4	$180\ast ones\left(size\right(t\left)\right)$	0	180	180
f5	$240-2.5t$	−2.5	240	237.5
f6	$90+1.5t$	+1.5	90	91.5

The results of the HODMD analysis are presented in Figure 4 and Figure 5.

Figure 4. Reconstruction of the signal after HODMD applied with the RMSE error.

Figure 5. HODMD results for the non-stationary signal (part 1).

The reconstructed signal in Figure 4 closely matches the original waveform, confirming the accuracy of the amplitude and phase extraction used in the synthesis. This non-stationary test case is challenging because it contains six distinct frequency components, spanning subsynchronous, fundamental, hypersynchronous, and harmonic ranges, that vary simultaneously, with a rate of change of frequency (RoCoF) between $1 \mathrm{Hz}$ and $2.5 \mathrm{Hz}$. Despite these dynamic conditions and the presence of a $50\%$ DC component, the HODMD spectrum analyzer accurately tracks all evolving phasors $f_1$ through $f_6$.

Table 4 shows that HODMD consistently delivers the lowest RMSE, confirming its superior capability to accurately track frequency trajectories even under significant ROCOF conditions (±2–2.5 Hz/s). In comparison, SST-STFT partially mitigates the bias from time–frequency resolution limits and ranks second in accuracy. ConceFT-SST, although robust to noise thanks to multi-taper averaging, produces broader spectral peaks that increase frequency bias, leading to the largest RMSE values.

Table 4. Comparison of RMSE(%) error of frequencies.

Frequency	HODMD	ConceFT-STFT	SST-STFT
30	$0.0195$	$0.02$	$0.02614$
60	$0.0179$	$0.1959$	$0.2549$
90	$2.301\times {10}^{-2}$	$0.97$	$0.0142$
120	$4.639\times {10}^{-3}$	$0.8165$	$0.01264$
180	$0.0222$	$1.4696$	$0.0824$
240	$0.028$	$0.3066$	$0.0425$

Figure 6a,b illustrate the time frequency representations obtained using the ConceFT-STFT and SST-STFT methods.

Figure 6. Time frequency representation results for the non-stationary signal using (a) ConceFT-STFT and (b) SST-STFT.

The ConceFT-STFT result (Figure 6a) exhibits sharp frequency localization and effectively highlights the dominant components with a smooth spectral energy distribution. However, a closer inspection shows that some interference patterns remain, particularly near the signal onset. These residual smears can hinder the isolation of weaker components in the presence of strong harmonics.

In contrast, the SST-STFT map (Figure 6b) provides a sparser representation, as expected from synchrosqueezing techniques. This enhances the visual clarity of the main components but also introduces high-frequency noise and vertical artifacts, particularly near the onset of the signal. These distortions, along with the method’s sensitivity to parameter selection, may explain the inconsistencies in spectral concentration.

The time–frequency map generated by HODMD (Figure 5) exhibits the most concentrated and well-structured representation among the three approaches. Each harmonic is sharply resolved as a distinct horizontal line with minimal spectral leakage, and the logarithmic amplitude scale confirms a consistent energy distribution across all six components, including those of lower amplitude.

Overall, SST-STFT improves sensitivity to time-varying signal components and offers clearer localization than standard STFT. However, for stationary or slowly varying signals, HODMD achieves superior accuracy in estimating frequencies, amplitudes, and phases, while avoiding the noise sensitivity and parameter dependence observed with SST-STFT.

4.2. Performance Evaluation on Laboratory-Scale Experimental Setup

4.2.1. Case Study 4: Measured Signal from a Battery Energy Storage System

In this case study, we evaluated the performance of HODMD, SST-STFT, and ConceFT-STFT on a current-measured signal obtained from a simulated battery system connected to a voltage source converter (VSC). The VSC converts the DC output of the bi-directional converter to AC, and the battery system output is connected to the grid through a substation. Voltage and current measurements at the point of common coupling (PCC) are monitored and provided to the proposed algorithm for harmonic identification. A schematic of the proposed system is shown in Figure 7.

Figure 7. Overview of the studied system.

Simulation Validation

The system was modeled in MATLAB with a detailed representation of the static converters, using a simulation time step of $10\mu \mathrm{s}$ at a fundamental frequency of $60 \mathrm{Hz}$. The sampling frequency was $25 \mathrm{kHz}$, with switching frequencies of $5 \mathrm{kHz}$ for the DC–DC converter and $2.7 \mathrm{kHz}$ for the DC–AC converter.

Figure 8 shows the original and reconstructed signals the corresponding frequency spectrum over a short time window. The two curves overlap almost perfectly. Even when zoomed in on a small segment, the reconstructed waveform closely follows the measured signal, demonstrating the ability of HODMD to capture the underlying dynamics with high fidelity. The residual error remains very small and does not distort the waveform shape, confirming the robustness of the method for accurate signal tracking.

Figure 8. HODMD results for the simulated system signal.

For a DC–AC converter modulated with sinusoidal pulse-width modulation (PWM), the frequency spectrum of the output voltage or current consists of the fundamental component at $f_1=60 \mathrm{Hz}$, permissible low-order harmonics (integer multiples of $f_1$), and high-frequency components clustered around the switching frequency $f_{\mathrm{sw}}=2700 \mathrm{Hz}$ and its odd multiples. The dominant high-frequency components can be expressed by the general relationship

$$f_h=\left|(2k+1)f_{\mathrm{sw}}\pm m{f}_1\right|,$$

(24)

where $k=0,1,2,\dots$ indexes the carrier harmonic groups and $m=1,2,3,\dots$ denotes the sideband order within each group. Based on the converter parameters, the theoretically expected harmonic frequencies are $60,120,540,1380,2280,5940,\dots \mathrm{Hz}$. As shown in the results, the HODMD algorithm successfully detects all these components. Therefore, HODMD is considered as the reference method for subsequent comparisons.

The results reported in Table 2 are consistent with those obtained for the simulated signal in Table 5. HODMD remains the most accurate method for stationary signal analysis. By contrast, the relatively large errors observed for SST-STFT and ConceFT-STFT indicate that these methods are less suitable for this class of signals.

Table 5. Frequency and amplitude identification for the measured system signal.

Method	freq (Hz)	Amp (pu)	Err freq (%)	Err Amp (%)
HODMD	20	0.00126	0
	60	1	0
	100	0.00499	0
	540	0.00118	0
	1380	0.00142	0
	2280	0.00131	0	As Reference
	5940	0.0024	0
	8340	0.0117	0
	9420	0.00199	0
	11,760	0.00189	0
STFT	27.2	0.018	35.67	1334
	67.45	1	12.413	0
	107.5	0.145	7.5	2797
	537.08	0.001	0.54	9
	1378.8	0.0012	0.083	15
	2279	0.0009	0.044	33
	5941	0.001	0.017	56
	8340	0.0033	0.061	72
	9,420	0.0004	0.054	18
	11,760	0.0016	0.063	13
ConceFT-STFT	18.6	0.1712	7	inf
	62.05	1	3.42	0
	101.3	0.525	1.3	inf
	540.6	0.008	0.12	584
	1381	0.0047	0.08	231
	2280	0.0014	0	10.3
	5941.4	0.001	0.02	58.3
	8340.4	0.0076	0.01	35.14
	9422	0.00189	0.02	38.35
	11,757.8	0.0018	0.02	6.17

Experimental Validation

The experimental validation was based on a laboratory-scale microgrid setup designed to replicate the realistic operating conditions of converter-dominated power systems. The OPAL-RT OP4510 real-time simulator served as the computational backbone of the experiments, providing the processing power needed for electromagnetic transient (EMT) simulations with high-level resolution. Converter control, data acquisition, and system monitoring were implemented through the HYPERSIM software interface, which ensures deterministic real-time execution and precise synchronization between the simulated power stage and the control algorithms.

This configuration enabled a full hardware-in-the-loop (HIL) implementation, offering a highly realistic testing environment that faithfully reproduces field conditions while avoiding the operational risks and costs of experiments on a live grid. The microgrid setup included a DC source emulating a battery energy storage system (BESS), a bi-directional DC–DC converter, and a grid-connected voltage source converter (VSC) with its filter. The laboratory-scale microgrid setup used for real-time validation of the proposed HODMD-based method is shown in Figure 9.

Figure 9. Laboratory-scale microgrid setup used for real-time validation of the proposed HODMD-based method.

The HODMD algorithm was applied to the measured voltage and current signals acquired at the point of common coupling (PCC) of the experimental setup. The resulting frequency–amplitude spectrum (Figure 10) clearly shows not only the fundamental component and low-order harmonics but also switching-related frequency components extending well beyond $40\mathrm{kHz}$. This highlights the capability of HODMD to capture both slow electromechanical oscillations and fast switching phenomena simultaneously, a key requirement for diagnosing converter-driven power systems.

Figure 10. HODMD results for the experimental setup signal.

These results confirm that HODMD can serve as a reliable tool for advanced harmonic analysis, event detection, and power quality monitoring in microgrids and large-scale networks. Its ability to operate on real-time measurement data makes it directly integrable into supervisory control and data acquisition (SCADA) systems or PMU-based wide-area monitoring frameworks, thus offering significant potential for practical deployment in grid operation centers.

4.2.2. Case Study 5: Frequency-Varying System Analysis

In this section, voltage and current measurement data were used to evaluate the performance of the three methods during a real event that occurred on the Hydro-Québec power grid in January 2001. This event was characterized by a significant frequency deviation, which was analyzed using a COMTRADE file recorded at a rate of 64 samples per cycle with a Phasor Measurement Unit (PMU) based on an Extended Kalman Filter (EKF) [27]. The fundamental frequency variation observed during this event, as measured by the PMU, is shown in Figure 11.

Figure 11. Frequency analysis during a grid event. (a) Frequency deviation represented by the PMU during the 2001 event [27]. (b) Comparison of grid frequency deviation estimation methods using HODMD, ConceFT-STFT, and SST-STFT.

The results shown in Figure 11b indicate that although all three methods capture and track the frequency changes, their performance differs considerably in terms of smoothness and time resolution. HODMD demonstrates superior accuracy, providing a precise, smooth, and noise-robust estimation of the fundamental frequency trajectory with minimal fluctuation, making it well suited for modeling and control applications. In contrast, ConceFT-STFT offers higher time–frequency resolution but shows significant sensitivity to noise, manifested as high-frequency fluctuations and transient spikes, particularly in the early time segments. SST-STFT provides an intermediate behavior, achieving improved resolution compared to HODMD while exhibiting lower noise sensitivity than ConceFT-STFT.

Unlike SST-STFT or ConceFT-STFT, which are designed to visualize the distribution of signal energy and amplitude in the time–frequency plane, HODMD extracts physically meaningful modal information such as frequency, damping, and phase. This makes the problem considerably more complex, since continuity must be ensured not only for frequency but also for damping and phase when using a sliding window. In our analyses of signals with sudden variations, such as the Hauer test signal with step changes and the Hydro-Québec signal exhibiting fundamental frequency excursions, we were able to maintain continuity of the fundamental mode’s frequency, damping factor, and phase across windows, despite edge effects. However, when many closely spaced modes are present simultaneously, mode switching (jumps) between consecutive windows may still occur, which is an inherent challenge of any parametric modal tracking method and requires careful post-processing.

5. Conclusions

In this paper, we investigated and benchmarked the performance of Higher-Order Dynamic Mode Decomposition (HODMD) against several advanced signal processing techniques across a range of applications. Through both synthetic and experimental analyses, HODMD demonstrated superior capability in accurately estimating the frequencies, amplitudes, and phases of multi-component signals. Nevertheless, it should be emphasized that HODMD is fundamentally a modal decomposition algorithm rather than a time–frequency analysis tool, which limits its effectiveness in scenarios requiring fine temporal localization of spectral features. In such cases, methods such as the Synchrosqueezing Transform-based Short-Time Fourier Transform (SST-STFT) provide enhanced sensitivity to time-varying components, while the Concentration of Frequency and Time (ConceFT-SST) approach offers improved frequency resolution compared with SST-STFT but shows limitations in reliably retrieving amplitude and phase information. Consequently, ConceFT-SST may be considered a complementary diagnostic tool for spectral characterization, whereas HODMD remains the preferred method for complete signal reconstruction and modal analysis. Also, as illustrated by the Hauer signal, HODMD can detect and track harmonic components with high fidelity during extreme events such as geomagnetic disturbances (GMDs), providing operators with valuable insight into harmonic propagation and interactions within the network and enabling preventive measures to reduce risks such as transformer saturation or protection misoperation. Together, these examples highlight how HODMD can be seamlessly integrated into existing PMU-based monitoring frameworks and Wide-Area Measurement Systems (WAMS), where its outputs can enhance control center analytics, improve dynamic model validation, and support decision-making tools such as oscillation source localization and wide-area damping control. Being fully data-driven, HODMD can be deployed without requiring intrusive interventions in the network, offering a practical, scalable, and robust solution to enhance situational awareness and grid resilience.