Stability Limit Analysis of DFIG Connected to Weak Grid in DC-Link Voltage Control Timescale

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The work presents an stability analysis of DFIG connected to weak grids and the proposed tools for this analysis. The analysis is particularized for a given system, so it is not clear if there is some conclusion that can be generalized to open the research to predict the stability limits i other scenarios. So, in the conclusion section, the values that the authors refer to seem to be related to the particular setup, it is not clear that those limits are general (SCR 1.3, for instance), so maybe that should be clearified in the conclusions section.

The proposed analysis is validated in both simulation and an experimental setup. 

It was difficult to follow the analysis at the beginning because the variables are only defined in an appendix. Furthermore, not always the nomenclature fits the figures or the appendix. For instance, Es Ir Udc Eg Ig Vt in Fig 1 then in the simplified equivalent in Fig 2 appears as Es Is Ic Ut Ug, but in equation (5) one can find the relationship between ic and is the split that is definen in the appendix. I don't know the reason to use E, V or U for the different voltages and how they are interchanged. I don't know if the appendix is mandatory, but, if there is enough space, maybe is helpful for the reader to present the variables when they are being used. 

In the experimental setup the pictures name Lf and Lg, but in figure 1 or 2, those L (or X) do not appear. 

Please, revise carefully anything related to the nomenclature and the coherence between figures, equations and appendix. 

 

 

 

 

Author Response

Comments 1: [The work presents an stability analysis of DFIG connected to weak grids and the proposed tools for this analysis. The analysis is particularized for a given system, so it is not clear if there is some conclusion that can be generalized to open the research to predict the stability limits i other scenarios. So, in the conclusion section, the values that the authors refer to seem to be related to the particular setup, it is not clear that those limits are general (SCR 1.3, for instance), so maybe that should be clearified in the conclusions section.]

Response 1: [Thank you for your valuable feedback. We agree that the stability limits discussed in the original conclusions were specific to our case study and may not be directly generalizable to all scenarios. To address this, we have revised the conclusions to clarify the context of the values and emphasize that these limits are derived from our specific setup while highlighting the broader methodological implications for other systems. We added a clarifying statement in the Conclusions section to explicitly mention that the stability limits are case-specific and depend on system parameters. The proposed analytical framework is generalizable and can be applied to predict stability limits in other scenarios by adjusting parameters like grid strength or control bandwidths. This change in the revised manuscript can be found in Page 16, Section 7 (Conclusions), Paragraph 1 and 5].

 

Comments 2: [The proposed analysis is validated in both simulation and an experimental setup. It was difficult to follow the analysis at the beginning because the variables are only defined in an appendix. Furthermore, not always the nomenclature fits the figures or the appendix. For instance, Es Ir Udc Eg Ig Vt in Fig 1 then in the simplified equivalent in Fig 2 appears as Es Is Ic Ut Ug, but in equation (5) one can find the relationship between ic and is the split that is definen in the appendix. I don't know the reason to use E, V or U for the different voltages and how they are interchanged. I don't know if the appendix is mandatory, but, if there is enough space, maybe is helpful for the reader to present the variables when they are being used.]

Response 2: [Thank you for your valuable feedback regarding the nomenclature consistency and variable definitions. We have made the following improvements to enhance clarity: We have standardized the voltage notation throughout the paper to consistently use "U" for voltages and "E" specifically for back-EMF, and we now introduce key variables when they first appear in the text rather than relying solely on the appendix. The simplified equivalent circuit in Figure 2 has been updated to match the nomenclature used in equations, and we have added explicit cross-references between figures, equations and variable definitions. These changes can be found in the revised manuscript in Page 3, Section 2.1, Paragraph 1, in Page 3, Section 2.2, Paragraph 1, in Page 4, Section 2.2, Paragraph 2, in Page 5, Section 2.3, Paragraph 3].

 

Comments 3: [In the experimental setup the pictures name Lf and Lg, but in figure 1 or 2, those L (or X) do not appear.].

Response 3: [ We are sorry for the mistake. Lf represents the inductance of filter. And Lg represents the inductance of transmission line. We add Lf and Lg to the Figure 1. Figure 2 is the simplified equivalent circuit diagram, so it only shows Xg. These changes can be found in the revised manuscript in Page 3, Section 2.1 and 2.2.]

 

Comments 4: [Please, revise carefully anything related to the nomenclature and the coherence between figures, equations and appendix.].

Response 4: [ We sincerely appreciate your careful review and valuable suggestions regarding the nomenclature consistency. We have thoroughly revised the manuscript to ensure complete coherence between all figures, equations, and the appendix. The changes include: (1) Standardizing all voltage notations to consistently use "U" for terminal/grid voltages and "E" specifically for back-EMF; (2) Verifying and aligning all variable definitions between the main text, figures, equations, and appendix; (3) Adding cross-references between equations and figures to clarify relationships; and (4) Including a notation guide footnote in Appendix A explaining our naming conventions. These improvements appear throughout the manuscript, particularly in Section 2 where we now explicitly define key variables before presenting the equivalent circuit and equations, and in all figure captions which now clearly link to the corresponding equation numbers. The appendix tables have also been updated to match the main text terminology exactly.]

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

electronics-3747553

Authors presented an analysis of stability limits of DFIG connected to a weak grid in DC-link voltage control timescale. Authors present the theory, simulation, and experimental results. My comments are as follows.

 

1. DFIG could be a common acronym in wind generators. Please spell it out at least once in the paper.

 

2. The variable Ug is used in many equations. In Appendix A, it is defined as 'Infinite bus voltage magnitude and phase'

It seems that Ug could not mean 'magnitude' and 'phase' at the same time. Is it a complex variable? Please check and make a correction if applicable.

 

3. Lamda_SCR in Equation 23 is one of the critical parameters in the analysis as it is used in Figure 10. However, Sac in Equation 23 is not defined in Appendix A. Please check.

 

4. Page 1: Authors wrote "However, in some areas, such as Gansu in China and Texas in USA, lots of wind power bases are located far away from load centers[4]-[6]. Transmitting large scales of wind power to load centers through long transmission lines will lead to wind turbines integrating to weak grid, which results in stability problems[7]-[9]."

1) Weak grid is one of the key constraints in the paper. Please define weak grid.

2) Is the weak grid the result of connecting wind turbines with a long transmission line? Or the grid is weak before connecting wind turbines?

 

5. Figures 10, 11, 12, and 13:

These figures show the stability trend versus parameter lambda_SCR, omega_AVC, and omega_PLL. Curves are plotted with parameter values in one side of the threshold value. Please consider adding curves with parameters in the other side of the threshold value for clarity. For example, please consider adding a curve for lambda_1.80 in Figure 10.

 

6. Page 13: Authors wrote "Fig. 15 depicts the experiment results with considering AVC in two cases: (a)

λSCR=1.22, (b) λSCR=1.20. And, Fig. 16 illustrates the experiment results without considering

AVC in two cases: (a) λSCR=1.20, (b) λSCR=1.19. Obviously, the results in Fig. 15 and Fig. 16

can well verify the stability analysis in Fig. 8."

1) Please explain how to set lamda SCR values in the experiment. Please explain it using the variables in Equation 23, i.e., U_g, U_to, and sin(theta_t0).

2) Please explain how to set omega_AVC and omega_PLL. Please explain how to set values for these parameters. Have you used a digital filter?

 

7. Figure 7: For the third curve from the top, check if the legend 'lambda_SCR = 1.2' is correct. It could be 1.5.

 

8. Figure 11: Caption "Response waveforms without considering AVC". Please check if 'without' needs to be corrected to 'with'.

 

9. Experimental results

1) It is good to include experimental results validating the proposed theory or algorithm. Please consider adding more explanation on the experimental setup.

2) Figures 16, 17, and 18 have been presented too briefly. Please consider adding some detailed analysis of experimental results.

In page 13, authors wrote "Fig. 15 depicts the experiment results with considering AVC in two cases: (a) λSCR=1.22, (b) λSCR=1.20. And, Fig. 16 illustrates the experiment results without considering AVC in two cases: (a) λSCR=1.20, (b) λSCR=1.19. Obviously, the results in Fig. 15 and Fig. 16 can well verify the stability analysis in Fig. 8."

Please consider analyzing Figures 15 and 16 with comparison to Figures 11 and 10, respectively.

In pages 13-14, authors wrote "Fig. 17 shows the experiment results with considering varied AVC bandwidths in three cases: (a) ωAVC=14.5Hz, (b) ωAVC=11Hz, (c) ωAVC=7Hz. And, Fig. 18 illustrates the experiment

results with considering varied PLL bandwidths in three cases: (a) ωPLL=3Hz, (b) ωPLL=13Hz, (c) ωPLL=26Hz. From the response of active power and DC-link voltage, it can be observed that the results in Fig. 17 and Fig. 18 can well verify the stability analysis in Fig. 9 and Fig. 10."

Please consider analyzing Figures 17 and 18 with comparison to Figures 12 and 13, respectively.

 

Comments on the Quality of English Language

English expressions needs to be improved in some places. In certain places, the meaning is somewhat unclear.

Author Response

Comments 1: [ DFIG could be a common acronym in wind generators. Please spell it out at least once in the paper. ]

Response 1: [ In response to the reviewer's suggestion, we have spelled out the acronym "DFIG" as "Doubly-Fed Induction Generator" at its first occurrence in the paper to ensure clarity for all readers. This change can be found on the first page, in the "Abstract" section, specifically in the first paragraph where the term is introduced. ]

 

Comments 2: [ The variable Ug is used in many equations. In Appendix A, it is defined as 'Infinite bus voltage magnitude and phase' It seems that Ug could not mean 'magnitude' and 'phase' at the same time. Is it a complex variable? Please check and make a correction if applicable. ]

Response 2: [ Thank you for catching this inconsistency. We have revised the definition of U_g in Appendix A to clarify that it represents only the magnitude of the infinite bus voltage. ]

 

Comments 3: [ Lamda_SCR in Equation 23 is one of the critical parameters in the analysis as it is used in Figure 10. However, Sac in Equation 23 is not defined in Appendix A. Please check. ]

Response 3: [ Thank you for identifying this omission. We have added the definition of Sac to Appendix A to ensure clarity, as it is essential for calculating λ_SCR in Equation (23). Sac represents the short-circuit capacity of the AC grid.]

 

Comments 4: [ Page 1: Authors wrote "However, in some areas, such as Gansu in China and Texas in USA, lots of wind power bases are located far away from load centers[4]-[6]. Transmitting large scales of wind power to load centers through long transmission lines will lead to wind turbines integrating to weak grid, which results in stability problems[7]-[9]." 1) Weak grid is one of the key constraints in the paper. Please define weak grid. 2) Is the weak grid the result of connecting wind turbines with a long transmission line? Or the grid is weak before connecting wind turbines? ]

Response 4: [In response to the reviewer's comment, we have clarified the definition of a weak grid as a power system with low short-circuit ratio, where grid strength is characterized by high impedance and limited power transfer capability, and we have specified that the weak grid condition arises primarily from the long transmission lines connecting wind turbines to load centers, rather than being inherent to the grid itself. This revision can be found on Page 1, Section 1, in the first paragraph starting with "However, in some areas...", where we added explanation to explicitly define weak grid and its relation to transmission line impedance, ensuring the context is clear for readers.]

 

Comments 5: [ Figures 10, 11, 12, and 13: These figures show the stability trend versus parameter lambda_SCR, omega_AVC, and omega_PLL. Curves are plotted with parameter values in one side of the threshold value. Please consider adding curves with parameters in the other side of the threshold value for clarity. For example, please consider adding a curve for lambda_1.80 in Figure 10. ]

Response 5: [In response to the reviewer's suggestion, we have updated Figures 10-13 by adding curves with parameter values on both sides of the threshold to clearly demonstrate the stability transition across critical values, which can be found in the revised manuscript on the respective figure pages.]

 

Comments 6: [ Page 13: Authors wrote "Fig. 15 depicts the experiment results with considering AVC in two cases: (a) λSCR=1.22, (b) λSCR=1.20. And, Fig. 16 illustrates the experiment results without considering AVC in two cases: (a) λSCR=1.20, (b) λSCR=1.19. Obviously, the results in Fig. 15 and Fig. 16 can well verify the stability analysis in Fig. 8." 1) Please explain how to set lamda SCR values in the experiment. Please explain it using the variables in Equation 23, i.e., U_g, U_to, and sin(theta_t0). 2) Please explain how to set omega_AVC and omega_PLL. Please explain how to set values for these parameters. Have you used a digital filter? ]

Response 6: [In response to the reviewer's comments, we have added detailed explanations in Section 6  clarifying that the λSCR values in experiments were set by adjusting the grid impedance Xg to vary Ug*Ug/(Xg*Pe) according to Equation 23, while ωAVC and ω_PLL were implemented through their respective PI controller gains in the FPGA-based control system without additional digital filters, with specific bandwidths calculated as ω_AVC=Ki_AVC and ω_PLL=√(Ki_PLL) to match the theoretical values in Table A1, and this operational detail is now explicitly stated in the revised manuscript's experimental setup description.]

 

Comments 7: [ Figure 7: For the third curve from the top, check if the legend 'lambda_SCR = 1.2' is correct. It could be 1.5. ]

Response 7: [We appreciate the reviewer's careful observation and confirm that the legend 'lambda_SCR = 1.2' was indeed incorrect. We have revised it to " lambda_SCR = 1.5".]

 

Comments 8: [ Figure 11: Caption "Response waveforms without considering AVC". Please check if 'without' needs to be corrected to 'with'. ]

Response 8: [We appreciate the reviewer's careful observation and confirm that the caption for Figure 11 was indeed incorrect; we have revised it from "Response waveforms without considering AVC" to "Response waveforms with considering AVC".]

 

Comments 9: [ It is good to include experimental results validating the proposed theory or algorithm. Please consider adding more explanation on the experimental setup. ]

Response 9: [In response to the reviewer's valuable suggestion, we have expanded the experimental setup description in Section 6 by adding detailed information about the 6kW DFIG test platform, including the configuration of the NI FPGA board 7868R and PXIe-1071 controller, the implementation of grid impedance Lg to emulate weak grid conditions, the specific measurement equipment used, the sampling rates for data acquisition, and the calibration procedures for ensuring measurement accuracy, with these additions appearing in the first two paragraphs of Section 6 where the experimental platform is initially introduced, providing readers with a clearer understanding of how the experimental validation was conducted.]

 

Comments 10: [ Figures 16, 17, and 18 have been presented too briefly. Please consider adding some detailed analysis of experimental results. ]

Response 10: [In response to the reviewer's constructive suggestion, we have significantly enhanced the analysis of experimental results in Figures 15-18 by adding detailed discussions as follows.

Experimental results in Fig. 15 demonstrate system behavior with AVC dynamics considered. Case (a) (λ_SCR=1.22) exhibits stable operation, characterized by well-damped oscillations converging to steady-state values in active power and DC-link voltage. In contrast, Case (b) (λ_SCR=1.20) shows clear instability, evidenced by sustained and growing oscillations in both P_e and U_dc. This validates the simulation findings in Fig. 11, confirming that the system becomes unstable below λ_SCR≈1.20 when AVC dynamics are active. Fig. 16 presents results without AVC dynamics. Case (a) (λ_SCR=1.20) remains stable, while Case (b) (λ_SCR=1.19) becomes unstable. This experimentally confirms the simulation results in Figure 10, demonstrating that excluding AVC dynamics raises the minimum stable SCR for Pe=1.0 p.u. from 1.20 with AVC to 1.22 without AVC, thus proving AVC dynamics improve the assessed stability limit under weak grid conditions.

Fig. 17 experimentally investigates the impact of varying AVC bandwidth under a fixed weak grid with λ_SCR=1.2. Case (a) (ω_AVC=14.5 Hz) maintains stable operation at the target Pe=1.0 p.u., with minimal Udc deviation. Cases (b) (ω_AVC=11 Hz) and (c) (ω_AVC=7 Hz), however, exhibit instability at active power levels significantly below 1.0 p.u. This directly verifies the simulation in Fig. 12, showing that a higher AVC bandwidth increases the dynamic stability limit. Fig. 18 tests PLL bandwidth effects under the same weak grid. Case (a) (ω_PLL=3 Hz) sustains stability at Pe=1.0 p.u. Cases (b) (ω_PLL=13 Hz) and (c) (ω_PLL=26 Hz) become unstable, with Udc exhibiting significant oscillations. This confirms the simulation results in Fig. 13, demonstrating that reducing PLL bandwidth enhances the stability limit under this specific weak grid condition with λ_SCR=1.2. The consistent response patterns between P_e and U_dc across all cases provide strong experimental validation for the theoretical bandwidth dependency analysis. This change in the revised manuscript can be found in Page 14, Section 6, Paragraph 2 and 3. ]

 

Comments 11: [ In page 13, authors wrote "Fig. 15 depicts the experiment results with considering AVC in two cases: (a) λSCR=1.22, (b) λSCR=1.20. And, Fig. 16 illustrates the experiment results without considering AVC in two cases: (a) λSCR=1.20, (b) λSCR=1.19. Obviously, the results in Fig. 15 and Fig. 16 can well verify the stability analysis in Fig. 8." Please consider analyzing Figures 15 and 16 with comparison to Figures 11 and 10, respectively. ]

Response 11: [In response to the reviewer's suggestion, we have enhanced the analysis in Section 6  by adding a detailed comparison between the experimental results and simulation results, specifically highlighting how the experimental waveforms for both AVC-included and AVC-excluded cases align with their corresponding simulation counterparts in terms of stability boundaries, transient response characteristics, and quantitative metrics. The results in Fig. 15 and Fig. 16 can well verify the stability analysis in Fig. 10 and Fig. 11.]

 

Comments 12: [ In pages 13-14, authors wrote "Fig. 17 shows the experiment results with considering varied AVC bandwidths in three cases: (a) ωAVC=14.5Hz, (b) ωAVC=11Hz, (c) ωAVC=7Hz. And, Fig. 18 illustrates the experiment results with considering varied PLL bandwidths in three cases: (a) ωPLL=3Hz, (b) ωPLL=13Hz, (c) ωPLL=26Hz. From the response of active power and DC-link voltage, it can be observed that the results in Fig. 17 and Fig. 18 can well verify the stability analysis in Fig. 9 and Fig. 10." Please consider analyzing Figures 17 and 18 with comparison to Figures 12 and 13, respectively. ]

Response 12: [In response to the reviewer's suggestion, we have expanded the analysis in Section 6 by adding a comprehensive comparison between the experimental results and simulation results, specifically examining how the experimental waveforms for varied AVC bandwidths and PLL bandwidths correlate with their simulation counterparts in terms of dynamic response characteristics, stability margins, and critical bandwidth thresholds. It can be observed that the results in Fig. 17 and Fig. 18 can well verify the stability analysis in Fig. 12 and Fig. 13.]

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

The main objective of the paper is to analyze the stability limits of a DFIG connected to a weak grid in the time interval of the DC voltage control, taking into account, mainly, the operating points and the dynamics of the control loops.

The problem identified by the authors is encountered in areas where there are long lines for transmitting electricity from the production area (wind turbine placement base) to the load centers (consumer locations).

The stability of a DFIG connected to a weak grid is strongly influenced by the dynamics of the control loops. It is interesting that - and it is the authors' merit - they achieve a hierarchy of the dynamics of these loops in the complex control system of the DFIG, on a multiple temporal scale.

In order to analyze the stability limits of the DFIG at the time scale of the DC voltage control (DVC), the authors propose a mathematical model of the integrated system that allows the detection of factors that can majorly influence its stability (in the case of connecting the DFIG to a weak network). This model represents a strong point of the work and represents - at the same time - an important contribution.

It is worth mentioning, in this context, the correlation - detected from the analysis of the mathematical model - between the mode of variation of the rotor current (along the d and q axes), the voltage at the stator terminals and the power developed by the DFIG.

The identification of this dynamic game - at a physical level - reflected by the model, allowed the authors to carry out a complex analysis of the static and dynamic stability of the DFIG connected to a weak network.

The key word - in the issue of stability - is that of "sensitivity". A sensitive system is one that reacts strongly to the output for small variations in the input quantities.

 As a strong point of the paper, it is worth mentioning, in this context, the identification of the major influence of the reactance of the network - in which the DFIG is connected - on the active power limit of the DFIG, but also on the short-circuit ratio of the system. The model established by the authors - under the simplifying assumption that the phase-locked loop (PLL) accurately tracks the phase of the terminal voltage - allows an analysis of the static stability of the analyzed system. It is worth noting - in this context - the role played by the Өt0 angle of the terminal voltage on the stability of the DFIG output power (this angle radically influences the short-circuit ratio of the system.

The issue of dynamic stability is much more complicated and the authors resort to visualizing the influence of the control loop dynamics on the stability limits of the DFIG connected to a weak network, using two scenarios, to detect the loop whose dynamics majorly influences the large-scale (dynamic) stability of the DFIG.

In short, reducing the PLL bandwidth or increasing the AVC bandwidth with constant damping can improve the stability limit of the DFIG system, taking into account the dynamics of all control loops (case/scenario II analyzed).

The conclusions obtained - based on the analysis performed, by taking into account the eigenvalues ​​- are correct, but, as they intuited authors, requires validation.

Validation came both through simulation and experiment.

As a result, the major contributions of the authors – in this work – can be summarized in the trinomial: mathematical model – validation on a virtual simulator – validation on a physical simulator.

It is remarkable that both the results obtained on a virtual simulator (based on a mathematical model implemented in Matlab/Simulink) and those obtained on a physical simulator (a specially designed experimental platform) validate the conclusions revealed in the analysis carried out, a priori, on the model proposed by the authors.

The conclusions formulated, at the end of the work, are fair and reflect the results obtained in the entire analysis in the presented work.

Observations and recommendations

1) The linearization of the equations of the proposed model was possible only for an analysis of the “small” stability of the system (what is called static stability, respectively, at small amplitude disturbing signals). The simplifying hypothesis under which this was done is mentioned later in the body of the paper, although it would have been good to specify it in this area of ​​the body of the paper, for the rigor of the approach).

2) The “large” stability, respectively, the dynamic stability of the system is analyzed in a multiscale temporal framework (from the order of 100 ms – alternating current control - to the order of 1 ms – rotor speed control - with the adjustment on the 10 ms DVC temporal scale - where the active power and terminal voltage control are found), well organized from a conceptual point of view, using the root locus method. The simulations performed also reflect the conclusions deduced as a result of using a rigorous scientific concept of stability analysis (root locus): according to figures 10, 11, 12 and 13 - the variation in time of the quantities Udc, Ut and Pe (in p.u.) is analyzed comparatively. In contrast, the experimentally obtained results no longer aim to reflect this whole but only Udc and Pe (in p.u.). Additionally, in figures 17 and 18 the temporary variation of the alternating current (AC) is presented. Moreover, in the experimental analysis it is stated that figures 17 and 18 reflect the stability analysis in figures 9 (conceptual analysis when varying the PLL bandwidth) and 10 (analysis on a virtual simulator in case AVC is not taken into account).

A more detailed explanation of what should really be compared would be necessary.

3) Figures 10 and 11 have the same explanation, although it is clear that the variations in the graphs are totally different. What is the truth?

Author Response

Comments 1: [The linearization of the equations of the proposed model was possible only for an analysis of the “small” stability of the system (what is called static stability, respectively, at small amplitude disturbing signals). The simplifying hypothesis under which this was done is mentioned later in the body of the paper, although it would have been good to specify it in this area of the body of the paper, for the rigor of the approach).]

Response 1: [ We appreciate the reviewer's valuable observation regarding the clarification of our linearization assumptions. We have modified Section 2.3 to explicitly state the small-signal stability analysis assumptions upfront for greater methodological rigor. The changes appear on page 5, Section 2.3, paragraph 1, where we clearly specify that: (1) The linearization is valid for small perturbations around an operating point, (2) The analysis assumes the system remains within a small neighborhood of equilibrium during disturbances, and (3) Higher-order terms are neglected in the Taylor series expansion. ]

 

Comments 2: [The “large” stability, respectively, the dynamic stability of the system is analyzed in a multiscale temporal framework (from the order of 100 ms – alternating current control - to the order of 1 ms – rotor speed control - with the adjustment on the 10 ms DVC temporal scale - where the active power and terminal voltage control are found), well organized from a conceptual point of view, using the root locus method. The simulations performed also reflect the conclusions deduced as a result of using a rigorous scientific concept of stability analysis (root locus): according to figures 10, 11, 12 and 13 - the variation in time of the quantities Udc, Ut and Pe (in p.u.) is analyzed comparatively. In contrast, the experimentally obtained results no longer aim to reflect this whole but only Udc and Pe (in p.u.). Additionally, in figures 17 and 18 the temporary variation of the alternating current (AC) is presented. Moreover, in the experimental analysis it is stated that figures 17 and 18 reflect the stability analysis in figures 9 (conceptual analysis when varying the PLL bandwidth) and 10 (analysis on a virtual simulator in case AVC is not taken into account). A more detailed explanation of what should really be compared would be necessary.]

Response 2: [ We thank the reviewer for this insightful observation regarding the experimental validation approach. We have enhanced the explanation of experimental-computational correlations in Section 6 with two key improvements: First, we explicitly state that the experimental results focus on Udc and Pe as primary stability indicators since these directly reflect the DC-link voltage control timescale dynamics that are the paper's focus, while AC current waveforms in Figs. 17-18 serve as secondary validation of PLL/AVC bandwidth effects. Second, in the captions of Figs. 15-18 we now specify which specific stability phenomena each experimental result verifies, with direct equation references. These changes create clearer correspondence between the multi-timescale theoretical analysis and selective but targeted experimental validations. ]

 

Comments 3: [Figures 10 and 11 have the same explanation, although it is clear that the variations in the graphs are totally different. What is the truth?]

Response 3: [ We sincerely apologize for the oversight in the original captions for Figures 10 and 11, which were mistakenly duplicated despite illustrating different experimental conditions. To rectify this, we have revised both captions to accurately reflect their distinct scenarios: Figure 10 clarifies that it depicts response waveforms without considering AVC dynamics, contrasting stable operation (λSCR=1.22) with instability onset (λSCR=1.20), while Figure 11 explicitly shows waveforms with AVC dynamics, demonstrating improved stability limits in alignment with Section 4’s root locus analysis. These corrections appear in the page 11, Section 5, paragraph 2, where we now explicitly differentiate the figures’ purposes and link them to their respective theoretical analyses. The figures themselves were also verified to ensure correct representation. ]

Author Response File: Author Response.pdf