# Assessment of Supraharmonic Emissions of Parallel Grid Connected Inverters Using Symmetric Carrier Phase Shift Concept

Abdellatif M Aboutaleb, Marwa S Osheba, Jan Desmet, Jos Knockaert

► To cite this version:

Abdellatif M Aboutaleb, Marwa S Osheba, Jan Desmet, Jos Knockaert. Assessment of Supraharmonic Emissions of Parallel Grid Connected Inverters Using Symmetric Carrier Phase Shift Concept. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0031. hal-05075750

HAL Id: hal-05075750

https://utc.hal.science/hal-05075750v1

Submitted on 20 May 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Assessment of Supraharmonic Emissions of Parallel Grid Connected Inverters Using Symmetric Carrier Phase Shift Concept

Abdellatif M. Aboutaleb

EELab/Lemcko
Ghent University
Kortrijk, Belgium

abdellatif.aboutaleb@ugent.be

Marwa S. Osheba

EELab/Lemcko
Ghent University
Kortrijk, Belgium

marwa.osheba@ugent.be

Jan Desmet

EELab/Lemcko
Ghent University
Kortrijk, Belgium

janj.desmet@ugent.be

Jos Knockaert

EELab/Lemcko
Ghent University
Kortrijk, Belgium

jos.knockaert@ugent.be

## Acknowledgements

This research is funded by Flanders Innovation & Entrepreneurship (VLAIO) and is part of the cSBO project IMPERFECT (HBC.2021.0175).

## Keywords

«supraharmonic», «grid connected inverter», « distributed power generation systems», «interleaved converters».

## Abstract

This paper presents an experimental study on the high frequency emissions in supraharmonic (SH) range of grid connected inverters (GCIs) under symmetric carrier phase shift concept. The aim of the study is to show the advantages and the limitations of using the carrier phase shift concept for parallel connected GCIs in distributed power generation systems. The experimental study is done with a dSPACE MicroLabBox based system of two GCIs under two symmetric and asymmetric scenarios.

## Introduction

The frequency range between 2 kHz and 150 kHz, known as the supraharmonic range [1-2], is still not completely covered by standards [3-4]. However, it attracts the interest of the researcher due to many reasons. First, the majority of grid connected converters (GCCs) are driven with switching rates that lie in the supraharmonic range [5]. Secondly, the behaviour of propagation and penetration of SHs is highly different from the harmonics below 2 kHz [2,6-7]. It is reported that, the SH components tend to propagate among the residential application instead of flowing in the grid [6]. This, in turn, causes electromagnetic interference (EMI) problems in such devices representing the third

reason why researchers are interested in this frequency range [8-13].

The conducted emissions in the SH range are divided into common mode and differential mode emissions [14]. It is reported that, in the SH range, the differential mode emissions are dominant, while the common mode emissions can be neglected. Exceptions are known for large systems, e.g. PV-plants and railway systems. Consequently, efforts have been done for deriving mathematical modelling approaches of grid connected converters by neglecting the effect of the parasitic components of such converters [15-16]. In [15], the differential mode noise in SH range have been modelled for the power factor correction (PFC) boost converter. The SH emission modelling approaches of single phase grid connected inverters (GCIs) has been derived in [16]. The two aforementioned studies have neglected the effect of the parasitic components effects on the total SH emissions of such converters.

The conducted SH emissions are divided mainly into two main types [1], primary emissions and secondary emissions. The first type is defined as the SH emissions measured at the converter terminals due to the switching behaviour of the converter itself. The secondary emissions are those SH emissions measured at the terminal of a converter due to the nearby converters.

Single-phase GCIs are considered one of the main sources of conducted SH emissions in the low voltage (LV) grids [16]. To reduce the high frequency (HF) emissions of GCI, Three solutions have been reported. The first one is the coupling filter design which faces many problems due to resonances and their dependency on the grid impedance [17-19]. The second solution is the carrier phase shift concept targeting specifically parallel-connected GCIs [20-21]. The third solution to mitigate the SH emissions of
GCIs is introduced in [22] by designing an interleaved active power filter for the GCIs.
In this paper, an experimental study is done to show the advantages of using the symmetric carrier phase-shift concept for parallel connected GCIs in distributed power generation systems. A dSPACE based system is implemented in the lab for two parallel GCIs. Two scenarios are studied, the first scenario is performed for symmetric GCIs while the second one is carried out for an asymmetric condition. In symmetric condition, the active power transfer, the coupling inductance value and the DC-link voltage are the same for all parallel GCIs. If one or more of the aforementioned parameters are asymmetric then, the parallel GCIs become asymmetric. In this paper the asymmetry in the coupling inductance value is considered for the second scenario.

## System description

The system under study is for two parallel GCIs. The system is investigated through simulation and experimental studies. The simulation study is done in MATLAB/Simulink. The schematic for the simulation model is as shown in Fig. 1. While the experimental study is performed on a setup that consists of a dSPACE MicroLabBox for controlling the GCIs. The GCIs are built using two Taraz variable frequency drives (VFDs) with the DC-link energized from the grid through a diode bridge rectifier (DBR). The power is transferred again to the grid with two legs of the VFD inverter forming a GCI. Since the DC-link voltage must be higher than the peak of the grid voltage, a step down transformer is used to decrease the grid voltage in the GCI side. Furthermore, the transformer gives the isolation needed since the grid works as a source as well as a load for the GCIs. The bandwidths of the current and voltage probes are 200 kHz and 100 kHz respectively. Fig. 2 shows an actual picture of the setup with the schematics of the power and control circuits.

A proportional resonance (PR) controller is used for each parallel GCI to control the error signal between the reference value of the grid side current and the measured one.

The data of the system currents (Fig. 2b) is captured by a 20 MHz PicoScope. The data is analysed by the means of the MATLAB software environment. The parameters of the system for the simulation and the experimental studies are as listed in Table I.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Schematic for the simulation model of the two parallel GCIs.</figcaption></figure>

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 2. dSPACE based setup for the experimental study. (a) Actual picture of the system. (b) System schematic. (c) PWM generation. (d) Control algorithm.</figcaption></figure>
<table><caption>Table I. System parameters for the simulation and the experimental studies.</caption><thead><tr><th>Parameter</th><th>Value</th></tr></thead><tbody><tr><td>DC-link voltage</td><td>325 V</td></tr><tr><td>Grid voltage (RMS)</td><td>66 V</td></tr><tr><td>Grid Frequency</td><td>50 Hz</td></tr><tr><td>Grid impedance</td><td>r<sub>g</sub> = 0.01Ω<br>L<sub>g</sub> = 0.5 mH</td></tr><tr><td>DC-link capacitors</td><td>470μF</td></tr><tr><td>Switching frequency</td><td>5kHz</td></tr><tr><td>Controller gains</td><td>K<sub>p</sub> = 0.8<br>K<sub>i</sub> = 234<br>K<sub>r</sub> = 1<br>B<sub>r</sub> = 1.5</td></tr></tbody></table>

## Scenario 1: Symmetric Conditions

The first scenario considers the symmetric case with two equal GCIs in parallel. The coupling inductance for each GCI is set at 20 mH. While the rest of the parameters are as in Table I. Two case studies are investigated for this scenario. The first case study is done for a symmetric condition of two parallel GCIs disregarding the carrier phase shift concept. The current results of this study are illustrated in Fig. 3 for the simulation study and Fig. 4 for the experimental study in both time and frequency domains.

<figure><img src="image_7.png" /><figcaption>Fig. 3. Simulated currents of two symmetric GCIs without carrier phase shift concept. (a) time domain, (b) frequency domain.</figcaption></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 4. Measured currents of two symmetric GCIs without carrier phase shift concept. (a) time domain, (b) frequency domain.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 5. Simulated currents of the two symmetric GCIs with carrier phase shift concept. (a) time domain, (b) frequency domain.</figcaption></figure>

It is manifest from the in Fig. 3 and Fig. 4 that, ignoring the carrier timing coordination between the two GCIs, is leading to SH emission aggregation in the total grid current at the point of common coupling. As seen from the time domain data in Fig. 3(a) and Fig. 4(a), the peak to peak ripple increases from about 1.3 A in each GCI current to about 2.8 A in the total current. The frequency domain data in Fig. 3(b) and Fig. 4(b) confirms the observations of the ones in the time domain. For example, the SH emission at 5 kHz increases in the total grid current to about 2 times its counterpart in each GCI current. This means, ignoring the control coordination between the
parallel GCIs leads to SH clustering at the point of common coupling (PCC).

<figure><img src="image_11.png" /><figcaption>Fig. 6. Measured currents of the two symmetric GCIs with carrier phase shift concept. (a) time domain, (b) frequency domain.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 7. Simulated currents of two asymmetric GCIs with carrier phase shift concept of scenario 2. (a) time domain, (b) frequency domain.</figcaption></figure>

Study 2 of scenario 1 takes into account the symmetric carrier phase shift concept between the two GCIs. Symmetric carrier phase shift means a time shift calculated from equation (1) is added among the parallel GCIs carrier signals.

$$T_{sym} = \frac{T_s}{N} \qquad (1)$$

Here $T_{sym}$ is the symmetric carrier phase shift time, $T_s$ is the switching period, and $N$ is the number of parallel converters. This means, for the

two GCIs with a 5 kHz switching frequency of study 2 of scenario 1, the added symmetric carrier phase shift is $10^{-4}$ second. The simulated and measured currents of both GCIs are given in Fig. 5 and Fig. 6 respectively.

<figure><img src="image_13.png" /><figcaption>Fig. 8. Measured currents of two asymmetric GCIs with carrier phase shift concept of scenario 2. (a) time domain, (b) frequency domain.</figcaption></figure>

It is evident that, the carrier phase-shift concept succeeds in reducing the total HF emissions of the parallel connected GCIs. Obviously, from the simulated time data (Fig. 5(a)) that, the peak to peak ripple in each GCI current is about 1.4 A while it is about 0.6 A in the total current. From the time domain data of the measured current (Fig. 6(a)), the peak to peak ripple in each GCI current is about 1.4 A while it is about 0.8 A in the total current. In comparison to the case without carrier phase shift, there are reductions of 58% and 43% of ripple current for the simulation and the measured data.

The results of frequency domain illustrated in Fig. 5(b) and Fig. 6(b) emphasise the SH emissions reduction of the parallel connected GCIs with symmetric carrier phase shift concept. It is apparent that, the SH bands with odd order decrease significantly. For example for the measured data, the SH at 5 kHz for each GCI is about 0.5 A. Without carrier shift, the amplitude at 5 kHz doubles to 1 A. With carrier shift, the 5 kHz component cancels out to 0.05 A. The reduction percentage in the 5kHz component with the carrier phase shift concept is about 95% from the counterpart of study 1.
## Scenario 2: Asymmetric Conditions

Asymmetry arises when one or more of the following parameters differ between the two parallel GCIs:

* The DC-link voltage.

* The coupling filter between the GCI and the grid.

* The active power transferred from the DC-link to the grid.

In the second scenario, asymmetry in the coupling inductance between the two parallel GCIs is considered. The coupling inductances of the first and second GCIs are set at 10 mH and 40 mH, respectively. The simulated and the measured results of scenario 2 are as depicted in Fig. 7 and Fig. 8 respectively. It can be seen from the results that the SH reduction under asymmetrical conditions is not at its optimal values from the frequency domain, the amplitudes at 5 kHz of the first GCI, the second GCI and the total current for the simulation results are at 0.96A, 0.2A and 0.96 A respectively and are at 1.08A, 0.2A and 1.08A respectively for the measured results.

## Discussion

Due to the pulse width modulation (PWM) driven nature of GCIs, multiple SH components are generated from the GCI and flow through the LV grid. The grid side current equation of one GCI is driven in [16] for sine-triangle modulation as:

$$i_g = i_{gf} + i_{gs} + i_{gsb} \quad (2)$$

Here, $i_g$ is the AC side current of the GCI, $i_{gf}$ is the fundamental component of the current, $i_{gs}$ are the baseband switching harmonics which are located at frequencies that are integer multiple of the grid frequency. The baseband switching harmonics are related to the switching frequency and they are not related to the grid frequency. and $i_{gsb}$ are the sideband switching harmonics of the GCI current which are related to the switching and the grid frequencies.

The three GCI current components can be represented as in equations (3), (4) and (5). In equations (3), (4) and (5). $V_{dc}$ is the DC voltage value, $M$ is the modulation index, $\omega_g$ is the grid angular frequency, $\delta$ is the load angle, $V_m$ is the amplitude grid voltage, $r$ is the coupling inductor resistance, $L$ is the coupling inductance, $J$ is the Bessel function of the first kind, $\omega_c$ is the switching angular frequency, $m \in \mathbb{N}_0$ is the order of harmonics generated from the carrier signal, $n \in \mathbb{N}_0$ is the order of harmonics generated by the reference signal within each carrier sideband harmonic, $\theta_0 = \tan^{-1}(\omega_g L / r)$ is the phase angle of the coupling filter inductance at $\omega_g$, $\theta_m = \tan^{-1}(m \omega_c L / r)$ is the phase angle of the coupling filter inductance at $m \omega_c$.

Equations (4) and (5) describe the conducted switching harmonics of GCI in SH range. It is clear from equation (4) that the baseband harmonics peaks depend on the coupling inductance value, the DC voltage value and the switching frequency. Additionally, it is obvious that the transferred power has no effect on the baseband harmonics peaks and phases. On the other hand, the phases of the sideband harmonics are dependent on the transferred power. However, the transferred power has no effect on the amplitudes of the switching harmonics. Table 1 summarize the parameters that affect the SH emissions of single phase GCIs.

As the parameters that mentioned in Table II affect the SH emissions of GCIs, the symmetric carrier phase shift concepts works efficiently for symmetric parallel connected GCIs. The symmetric carrier phase concept depends on two facts: SH emissions of N symmetric GCIs are almost identical with the same amplitudes. The phase of these harmonics depends on the phase of the carrier if the transferred powers are the same. Consequently, applying the symmetric carrier phase shift concept leads to making the switching harmonics bands of GCI to form an N-phase system. As a result of that, the summation of the

$$i_{gf} = \frac{V_{dc} M \cos(\omega_g t + \delta - \theta_0) - V_m \cos(\omega_g t - \theta_0)}{\sqrt{(r^2 + (\omega_g L)^2)}} \quad (3)$$

$$i_{gs} = \frac{4V_{dc}}{\pi} \sum_{m=1}^{\infty} \left[ \frac{J_0 m \pi M}{2m^2 \omega_c L} \sin\left(m \frac{\pi}{2}\right) \cos(m \omega_c t - \theta_m) \right] \quad (4)$$

$$i_{gsb} = \frac{4V_{dc}}{\pi} \sum_{m=1}^{\infty} \sum_{\substack{n=-\infty \\ n \neq 0}}^{\infty} \left[ \frac{J_n m \pi M}{2m^2 \omega_c L} \sin\left(m + n\right) \frac{\pi}{2} \right] \cos\left(m \omega_c t + n \delta - \frac{\pi}{2}\right) \quad (5)$$
first N-1 bands at the PCC becomes zero. Just only the band with order N and its integer multiples appears in the frequency spectra of the total current. For instance, when three symmetric parallel GCIs are controlled using the carrier phase shift concept, the sum of the first, second, fourth, fifth, seventh, etc., switching harmonic bands cancels out at the PCC. Consequently, the frequency spectrum of the total current includes only the third, sixth, ninth, etc., of the switching harmonics. In the scenarios discussed in the paper, the number of symmetric GCIs is two. Under symmetric conditions, only the second, fourth, sixth, etc., switching harmonic bands appear in the frequency spectrum of the total current.

<table><caption>Table II. Parameters that affect the emissions of single-phase GCIs in SH range.</caption><thead><tr><th></th><th>Baseband<br>Harmonics</th><th>Sideband<br>harmonics</th></tr></thead><tbody><tr><th>Amplitude</th><td><ul><li>DC-link voltage</li><li>Coupling filter inductance</li><li>Modulation ratio</li><li>Switching frequency</li></ul></td><td></td></tr><tr><th>phase</th><td><ul><li>Carrier<br>phase</li></ul></td><td><ul><li>Carrier phase</li><li>Load angle<br>(Active Power<br>transferred)</li></ul></td></tr></tbody></table>

## Conclusion

Two parallel connected GCIs with a symmetric carrier phase shift have been studied by experiment. The carrier phase shift is normally used to reduce the total SH emissions to the grid from a system of parallel GCIs. It has been concluded from the experimental study that;

* The symmetric carrier phase shift is optimal to use for parallel symmetric GCIs to reduce the HF emissions in the SH range.

* However, for parallel asymmetric GCIs, the symmetric carrier phase shift filtering performance decreases in addition to the need for a centralized controller that distributes the same carrier signal to the controllers of the GCIs system.

## References

[1]. J. Meyer et al., "Future work on harmonics - some expert opinions Part II - supraharmonics,

standards and measurements," 2014 16th International Conference on Harmonics and Quality of Power (ICHQP), Bucharest, Romania, 2014, pp. 909-913, doi: 10.1109/ICHQP.2014.6842871.

[2]. Á. Espín-Delgado and S. Rönnberg, "Modeling and Analysis of Supraharmonic Propagation for Stochastic Studies," in IEEE Transactions on Power Delivery, vol. 37, no. 6, pp. 4899-4910, Dec. 2022, doi: 10.1109/TPWRD.2022.3162712.

[3]. S. T. Y. Alfalahi et al., "Supraharmonics in Power Grid: Identification, Standards, and Measurement Techniques," in IEEE Access, vol. 9, pp. 103677-103690, 2021, doi: 10.1109/ACCESS.2021.3099013.

[4]. A. A. Alkahtani et al., "Power Quality in Microgrids Including Supraharmonics: Issues, Standards, and Mitigations," in IEEE Access, vol. 8, pp. 127104-127122, 2020, doi: 10.1109/ACCESS.2020.3008042.

[5]. Espín-Delgado, Ángela, Tatiano Busatto, Vineetha Ravindran, Sarah K. Rönnberg, and Jan Meyer. "Evaluation of supraharmonic propagation in LV networks based on the impedance changes created by household devices." In 2020 IEEE PES Innovative Smart Grid Technologies Europe (ISGT-Europe), pp. 754-758. IEEE, 2020.

[6]. A. Espin Delgado and S. Rönnberg, 'Assessment of Grid Impedance Impact On Supraharmonic Propagation', in CIRED 2021 - The 26th International Conference and Exhibition on Electricity Distribution, 2021, pp. 688-692.

[7]. A. M. Aboutaleb, G. G. Roumeliotis, J. Desmet and J. Knockaert, "Analysis of transmission line modelling in the MATLAB/Simscape software package," 27th International Conference on Electricity Distribution (CIRED 2023), Rome, Italy, 2023, pp. 4009-4013, doi: 10.1049/icp.2023.0493.

[8]. S. Sakar, S. Rönnberg and M. Bollen, "Interferences in AC-DC LED Drivers Exposed to Voltage Disturbances in the Frequency Range 2-150 kHz," in IEEE Transactions on Power Electronics, vol. 34, no. 11, pp. 11171-11181, Nov. 2019, doi: 10.1109/TPEL.2019.2899176.

[9]. S. Sakar and S. Rönnberg, "Modeling and Analysis of DC-Link Capacitors Subjected to High Frequency Conducted Disturbances in Electronic Equipment," in IEEE Transactions on Power Electronics, vol. 37, no. 5, pp. 5949-5956, May 2022, doi: 10.1109/TPEL.2021.3121883.

[10]. S. Sakar, S. K. Rönnberg and M. Bollen, "Interharmonic Emission in AC-DC Converters
Exposed to Nonsynchronized High-Frequency Voltage Above 2 kHz," in IEEE Transactions on Power Electronics, vol. 36, no. 7, pp. 7705-7715, July 2021, doi: 10.1109/TPEL.2020.3047862.

[11]. A. M. Aboutaleb, A. Gholizad, J. Desmet and J. Knockaert, "Self-Induced and Mutual-Induced Emissions of Grid-Tied Converters," in IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 12, no. 4, pp. 3875-3886, Aug. 2024, doi: 10.1109/JESTPE.2024.3401255.

[12]. A. M. Aboutaleb, M. S. Osheba, J. Desmet, and J. Knockaert, "Comparison of the Impacts of Background Distortion with Even and Odd Orders in the Supraharmonic Range on Grid-Connected Converters," Electronics, vol. 13, no. 20, p. 4023, Oct. 2024, doi: 10.3390/electronics13204023.

[13]. M. S. Osheba, A. M. Aboutaleb, J. Desmet, and Jos Knockaert, "The Impact of Grid Distortion on the Power Conversion Harmonics of AC/DC Converters in the Supraharmonic Range," Electronics, vol. 13, no. 12, pp. 2244-2248, Jun. 2024, doi: https://doi.org/10.3390/electronics13122244.

[14]. Sandrolini, Leonardo, and Andrea Mariscotti. "Waveform and spectral characteristics of supraharmonic unsymmetrical conducted EMI of switched-mode power supplies." Electronics 11, no. 4 (2022): 591.

[15]. N. Nourani Esfetanaj, H. Wang, F. Blaabjerg and P. Davari, "Differential Mode Noise Prediction and Analysis in Single-Phase Boost PFC for the New Frequency Range of 9-150 kHz," in IEEE Journal of Emerging and Selected Topics in Industrial Electronics, vol. 3, no. 1, pp. 177-187, Jan. 2022, doi: 10.1109/JESTIE.2021.3066320.

[16]. A. Aboutaleb, J. Desmet, and J. Knockaert, "Impact of grid-connected inverter parameters on the supraharmonic emissions in distributed power generation systems," MACHINES, vol. 11, no. 11, 2023.

[17]. P. Alemi, C. -J. Bae and D. -C. Lee, "Resonance Suppression Based on PR Control for Single-Phase Grid-Connected Inverters With LLCL Filters," in IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 4, no. 2, pp. 459-467, June 2016, doi: 10.1109/JESTPE.2015.2464699.

[18]. L. Liu, J. Xu, J. Ye and A. Shen, "Passivity Enhancement and Grid-Current Distortion Mitigation for Inverter-Side Current Controlled LCL-Type Grid-Connected Inverters," in IEEE Access, vol. 12, pp. 66505-66515, 2024, doi: 10.1109/ACCESS.2024.3396923.

[19]. K. Koiwa, H. Takahashi, T. Zanma, K. -Z. Liu, K. Natori and Y. Sato, "A Novel Filter With High Harmonics Attenuation and Small Dimension for Grid-Connected Inverter," in IEEE Transactions on Power Electronics, vol. 38, no. 2, pp. 2202-2214, Feb. 2023, doi: 10.1109/TPEL.2022.3207432.

[20]. J. Poon, B. Johnson, S. V. Dhople and J. Rivas-Davila, "Decentralized Carrier Phase Shifting for Optimal Harmonic Minimization in Asymmetric Parallel-Connected Inverters," in IEEE Transactions on Power Electronics, vol. 36, no. 5, pp. 5915-5925, May 2021, doi: 10.1109/TPEL.2020.3030009.

[21]. S. Baburajan, H. Wang, D. Kumar, Q. Wang, and F. Blaabjerg, "DC-Link current harmonic mitigation via Phase-Shifting of carrier waves in paralleled inverter systems," Energies, Perret F.: Development of GaN diodes for high voltage applications, EPE Journal Vol. 19 no 6 pp 63-72.

[22]. A. M. Aboutaleb, J. Desmet and J. Knockaert, "A Novel Active Power Filter for Supraharmonic Emissions of Single Phase Grid-Connected Inverters," in *IEEE Transactions on Power Electronics*, doi: 10.1109/TPEL.2025.3539024.