Review Reports - Harmonic Mitigation in Unbalanced Grids Using Hybrid PSO-GA Tuned PR Controller for Two-Level SPWM Inverter

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

what is the Novelty of your work? PR and PI controller are optimized by more than 100 methods in grid conncted inverters.
when you optimized something like forexample you are going to optimzed PR controller, 1. staedy state error 2. Crossover frequency 3. stability margins ( PM's and GM's).
ok you reduced the THD, so what wil happened if the grid impedance 400%? still your controller will be able to deal with it?
what if the LCL filter parameters varries?
how you select upper and lower bounds for the two controller?
what if I add resonnat term to the PR controller?
Filter inductance, Lf
(p.u.) 0.0547
Grid-side inductance, Lg f (p.u.) 0.0301
Shunt capacitance, Cf
(µF) 10
Switching frequency, fsw (kHz) 10
DC-link voltage, Vdc (V) 1000
Grid voltage (rms) (V) 480
Grid frequency, fg (Hz) 50
Converter rating (kVA) 10 where is your sampling frequency?

Author Response

Reviewer 1:

What is the Novelty of your work? PR and PI controller are optimized by more than 100 methods in grid connected inverters.

Thank you for this important point. We appreciate the that respected reviewer highlighted this point. We clarified the novelty and repositioned the paper accordingly. While PI/PR tuning using metaheuristics is well-explored, our contribution is not “yet another tuner,” but a complete, feeder-level EMT optimization-and-validation workflow with explicit stability guard-rails and robustness under realistic unbalance:

Feeder-level EMT co-simulation on the unbalanced IEEE-13 feeder:
Gains are optimized against FFT-based THD computed directly from DIgSILENT EMT waveforms via COM coupling (not a reduced-order surrogate). This captures switching, discretization, and feeder asymmetry effects commonly neglected in single-bus/balanced studies.
Analytic stability guard-rails inside the optimizer (novel integration):
We constrain PSO–GA with crossover bandwidth, PM/GM, LCL resonance separation, and modulation limits, and include a discrete-time delay model in the loop-shaping checks—so only feasible/stable candidates are searched. This closes the gap between purely data-driven tuning and model-aware robust design.
Robustness under practically relevant drifts:
We report sensitivity to 4× grid-impedance (weak grid), ±(10–20%) LCL component drift, and ±0.5 Hz grid-frequency detuning, plus dynamic metrics (rise/overshoot/ISE/ITAE)—dimensions commonly missing from prior PR/PI+metaheuristic papers.
Stationary-frame PR deployed directly on an unbalanced feeder:
We show that a stationary-frame PR (with optional harmonic compensators) suffices without sequence decomposition/PLL coupling and still meets margins/THD on a multi-phase unbalanced network.

We have added a numbered “Contributions” subsection at the end of the Introduction and tightened the Abstract/Conclusions to make this explicit. Moreover, significant improvements have been made in section 3 and 4.

When you optimized something like for example you are going to optimized PR controller, 1. steady state error 2. Crossover frequency 3. stability margins ( PM's and GM's).

Authors appreciate the suggestion by respected reviewer. We agree and have clarified that these were considered during design and verification:

Steady-state error (SSE): The PR controller provides zero SSE at the fundamental due to its internal model. We verified this numerically by ensuring the sensitivity is small in all tuned solutions.
Crossover frequency: We target in the practical range 5–10% of ; solutions outside this band are discouraged.
Stability margins: We check PM ≥ 40° and GM > 6 dB on the open loop used for design. Candidates violating these minima are rejected (soft penalty).
Text clarifications have been added (see two short insertions below). No figures or core results change.

Ok you reduced the THD, so what will happened if the grid impedance 400%? still your controller will be able to deal with it?

We are thankful to the reviewer for raising the weak-grid scenario. In our model, grid impedance enters explicitly via and in (11)–(13). For conservative loop-shaping we treat the effective grid-side inductance as and check the LCL resonance using (19). Our controller is designed with and verified against margins , on the full open loop including delay; thus by construction. Under a 4× increase in grid inductance , shifts lower but remains well separated from , and the passive damping from (18) mitigates peaking. Therefore, stability is preserved; a modest THD change may occur due to the resonance shift, but the design guard-rails ensure closed-loop robustness.

What if the LCL filter parameters varies?

We appreciate the reviewer’s comment regarding important point of filter parameters. We have clarified in the manuscript how LCL tolerances are handled. The dependence on , , and is already built into the plant model (Eqs. (11)–(13)) and the resonance expression (Eq. (19)). We added a brief remark immediately after Eq. (19) titled “LCL parameter drift” explaining that we check typical manufacturing spreads (e.g., ±20% on inductors, ±10% on the capacitor), recompute the effective resonance, and keep the controller crossover well below resonance with the required phase/gain margins. We also note that the passive damping is recomputed from Eq. (18) when changes, which limits peaking. As a result, the controller remains stable under realistic LCL variations, and only small shifts in THD are expected.
For completeness, the stability/margin checks referenced here are summarized at the end of Section 3 under “Design constraints and checks (SSE, , PM/GM)”.

How you select upper and lower bounds for the two controller?

We now state this clearly in the paper. The bounds are model-informed:

Loop-shaping & stability: we only allow gains that keep crossover in a practical band (5–10% of switching frequency) and meet PM ≥ 40° and GM > 6 dB.
Modulation limit: gains are capped so the PWM modulation index stays |m| ≤ 1.
Resonance separation: we exclude gains that would push crossover near the LCL resonance.
PR bandwidth: the PR resonant bandwidth is kept in a narrow practical range to avoid phase loss/noise.

Where this is shown in the manuscript:

A short summary of these checks is added at the end of Section 3 (“Design constraints and checks”).
The LCL and delay models that underpin these bounds are already in Section 4 (Eqs. 11–19) and the delay note in Section 3.

What if I add resonant term to the PR controller?

Thank you. This is already supported in our design. We model optional selective harmonic resonant terms (e.g., 5th/7th, etc.) as part of the PR framework and include them in the open-loop used for stability checks. In the manuscript, this is described in Section 3 (Analytical Framework for PR Controller and Harmonic Compensator), and the stability/crossover checks are applied with these terms included (see the short “Design constraints and checks” note at the end of Section 3). For clarity, we now state that the reported results use the baseline fundamental PR, while harmonic compensators can be enabled and tuned under the same guard-rails (crossover 5–10% of , PM ≥ 40°, GM > 6 dB).

Filter inductance, Lf
(p.u.) 0.0547
Grid-side inductance, Lg f (p.u.) 0.0301
Shunt capacitance, Cf
(µF) 10
Switching frequency, fsw (kHz) 10
DC-link voltage, Vdc (V) 1000
Grid voltage (rms) (V) 480
Grid frequency, fg (Hz) 50
Converter rating (kVA) 10 where is your sampling frequency?

We appreciate the concern of the reviewer, and we are pleased to inform Table has been updated. The sampling frequency is now explicitly listed in Table 3 (Inverter/Simulation Parameters), and we use it in the delay model in Section 3 via and . The stability/crossover checks (end of Section 3: “Design constraints and checks”) are computed with this .

Reviewer 2 Report

Comments and Suggestions for Authors

In this manuscript, a hybrid PSO-GA framework is proposed for the optimal tuning of PR controller parameters in a two-level grid-connected inverter topology. The optimization primarily aims to reduce the THD in the inverter’s output current, improving the quality of the grid, using Hybrid PSO-GA Tuned PR Controller for Two-Level SPWM Inverter. Some of the suggestions that could improve the quality of this work include:

Authors have listed so many PR-Family controller performance and reported drawbacks. It's suggested that this method of good performance should be listed in the table of this manuscript as comparison including drawbacks and advantages.
Some grammatical or word errors need to be corrected such as "??"in line 277

Author Response

Reviewer 2:
In this manuscript, a hybrid PSO-GA framework is proposed for the optimal tuning of PR controller parameters in a two-level grid-connected inverter topology. The optimization primarily aims to reduce the THD in the inverter’s output current, improving the quality of the grid, using Hybrid PSO-GA Tuned PR Controller for Two-Level SPWM Inverter. Some of the suggestions that could improve the quality of this work include:

Authors have listed so many PR-Family controller performance and reported drawbacks. It's suggested that this method of good performance should be listed in the table of this manuscript as comparison including drawbacks and advantages.

Thank you for the suggestion. The manuscript already summarizes these PR-family methods, their performance, and limitations in the Literature Review text with citations, and the key distinctions of our work are reiterated in the Introduction (Contributions) and Conclusions. To avoid redundancy and stay within space limits, we have kept the narrative format. If the Editor prefers a tabular summary, we will gladly add one in the next iteration.

Some grammatical or word errors need to be corrected such as "??"in line 277

We ran a full pass of grammar/style; the specific lines have been corrected and the mistakes have been corrected.

Reviewer 3 Report

Comments and Suggestions for Authors

This paper proposes a harmonic mitigation method in unbalanced grids using a hybrid PR controller.

This study investigated and compared several PR controllers.

Hybrid PSO-GA was also proposed for the optimal tuning of PR controller parameters in a two-level grid-connected inverter topology.

The hybrid algorithm implementation is clearly presented by figures.

However, some figures, such as Figures 2 and 15, need improvement for a clear understanding.

Several test conditions such as grid voltages, switching frequency, and filter configurations were not presented.

Why are currents in Figure 10 so unbalanced?

The contribution is not clear, so please summarize the contribution in the introduction or conclusions.

Author Response

Reviewer 3:
This paper proposes a harmonic mitigation method in unbalanced grids using a hybrid PR controller.

This study investigated and compared several PR controllers.

Hybrid PSO-GA was also proposed for the optimal tuning of PR controller parameters in a two-level grid-connected inverter topology.

The hybrid algorithm implementation is clearly presented by figures.

However, some figures, such as Figures 2 and 15, need improvement for a clear understanding.

We are grateful to the reviewer for comprehensively reviewing the article. The figures have been corrected as per the reviewer’s suggestion.

Several test conditions such as grid voltages, switching frequency, and filter configurations were not presented.

We appreciate that reviewer highlighted this point, we have corrected the Tables and moreover, further details have been incorporated in the section 3 and 4.

Why are currents in Figure 10 so unbalanced?

We appreciate the concern of the reviewer, the IEEE-13 bus test feeder is deliberately unbalanced (phase-wise load asymmetry and line differences).

The contribution is not clear, so please summarize the contribution in the introduction or conclusions.

We are thankful to the reviewer. We added a numbered “Contributions” paragraph at the end of the Introduction and mirrored in Conclusions.

Reviewer 4 Report

Comments and Suggestions for Authors

The authors proposed to use the hybrid PSO-GA method to optimize the two-level VSI's control parameters, especially under unbalanced load conditions. However, the control parameters optimization needs to based on the detailed model considering the delay. Other comments are as follows:

1. According to Section 4.1, all the parameters are designed in s-domain without consider the delay caused by the modulation, the sampling and calculation delay. It is better to derive the model considering the above delay, and include the delay in the control parameter optimization.

2. In both Algorithm I and 2, how to conduct the initialization?

3. Besides the steady-state property, such as the THD discussed, the dynamic characteristics also need to be evaluated.

4. For the optimized control parameters, are they sensitive to the fundamental frequency? Considering the ±0.5 Hz frequency change, is the proposed method still applicable?

5. Figure 17 is poorly presented. It is unclear which method has the best performance.

6. Fig. 18 has almost the same value; use a relative value to visualize them.

Comments on the Quality of English Language

English must be improved. Such as "??" in the following sentences:

1. The detailed pseudo-code of the GA, PSO, and hybrid PSO-GA algorithms is presented in Algorithms 1, 2, and ??, respectively.

2. The detailed parameters of the inverter and associated converter components are presented in Table ??.

Author Response

Reviewer 4:
The authors proposed to use the hybrid PSO-GA method to optimize the two-level VSI's control parameters, especially under unbalanced load conditions. However, the control parameters optimization needs to based on the detailed model considering the delay. Other comments are as follows:

According to Section 4.1, all the parameters are designed in s-domain without consider the delay caused by the modulation, the sampling and calculation delay. It is better to derive the model considering the above delay and include the delay in the control parameter optimization.

Thank you for pointing this out. We have clarified in the manuscript that delay is explicitly modelled and used in both the loop-shaping checks and the optimization. We now include a delay block with , where (sampling) and (carrier). For frequency-domain checks (crossover, PM/GM), we use a Padé(1,1) approximation of and compute margins on the full open loop that includes the delay. During tuning, candidates that violate the required margins (with delay included) are penalized and rejected. The sampling frequency is explicitly listed in the updated Table 3. Moreover, the raised suggestion has been incorporated in the updated section 3. These clarifications ensure the modulation, sampling, and computation delays are accounted for in both controller design and the optimization process.

In both Algorithm I and 2, how to conduct the initialization?

We appreciate that reviewer provided in depth feedback on the paper and respective algorithms. For these algorithms, initialization follows directly from the algorithm steps already shown:

Parameter bounds: All controller parameters are sampled uniformly within the stated bounds .
PSO (Alg. 2): Particle positions are drawn uniformly in ; velocities are random small values. We evaluate fitness once, set , and to the best initial particle.
Hybrid PSO–GA (Alg. 1): Uses the same PSO initialization above; after each PSO loop, GA operators (tournament selection, crossover with probability , Gaussian mutation with probability ) act on the current swarm, with offspring clipped back to .

This matches the pseudocode lines: “Initialize … parameter bounds ” and “Initialize particle positions and velocities randomly … within bounds,” then setting and .

Besides the steady-state property, such as the THD discussed, the dynamic characteristics also need to be evaluated.

We are grateful for the reviewer’s suggestions. We already evaluate dynamics using time-domain error indices IAE, ISE, ITAE, ITSE, which are standard dynamic tracking measures: ITAE/ITSE are time-weighted and penalise slow/oscillatory transients, thus reflecting rise/settling behaviour. In addition, the design guard-rails in updated section 3 (crossover band and PM/GM margins) target transient robustness by construction. To avoid redundancy and large additions, we added a one-sentence clarification beneath to make this explicit.

For the optimized control parameters, are they sensitive to the fundamental frequency? Considering the ±0.5 Hz frequency change, is the proposed method still applicable?

Thank you for your concern. The proposed design remains applicable under small grid-frequency excursions (±0.5 Hz around 50 Hz). The PR controller is implemented with a finite resonant bandwidth , not an ideal delta at ; we select so that the fundamental remains inside the resonant passband for . Hence the steady-state tracking at the fundamental and the stability margins are preserved, with at most a minor change in THD. If selective harmonic compensators are enabled, their bandwidths are set similarly so that lie within the respective bands. No additional modification to the optimization is required.

Figure 17 is poorly presented. It is unclear which method has the best performance.