Physics-informed GNN for non-linear constrained optimization: PINCO, a solver for the AC-optimal power flow

We appreciate the detailed feedback provided; while some concerns highlight valid areas for clarification and improvement, we believe other critiques may stem from misunderstandings or misalignment of expectations regarding the goals of our work. Our focus is on solving the AC-OPF problem in a fully unsupervised manner using a combined GNN+PINN framework, differentiating this work from others relying on supervised learning techniques. We believe this addresses many of the comments regarding comparisons with prior works. While we have decided not to proceed further with the submission, we would still like to take this opportunity to clarify the points raised.

1. While the referenced works address multiple generators per bus, they rely on supervised learning approaches or tackle entirely different problems (e.g., unit commitment). Our approach, by contrast, solves the AC-OPF problem in a fully unsupervised manner. Additionally, we stress that aggregating and disaggregating generators, as suggested by the reviewer, introduce inaccuracies. In practical settings, operators require detailed generator-level outputs—not aggregated approximations. Particularly, how do we deal with generators but with different costs?

2. We disagree with this claim and invite the reviewer to refer to the references [1] [2]. Both references provide theoretical backing for the Augmented Lagrangian approach employed in our work. We want to ask if you could elaborate on why achieving inequality constraints equal to zero is not theoretically sound. [1] Lu et Al., "Physics-Informed Neural Networks with Hard Constraints for Inverse Design" [2] Toussaint M., "Introduction to Optimization"

3. We acknowledge the lack of experiments on this aspect. In the next version of the manuscript, we will include a case study demonstrating the proposed model's ability to handle topology variations.

4. While we agree that enforcing inequality constraints can sometimes be straightforward, we emphasize that prior fully unsupervised approaches have not achieved this in the AC-OPF context. Our work demonstrates zero inequality violations in a fully unsupervised setting, a non-trivial achievement. Additionally, the results on equality constraint violations are comparable to traditional solvers, as reported in Tables 1 and 2.

5. We disagree for several reasons:

Realism of Test Systems:

• Many real-world transmission systems, such as the Swiss transmission grid, comprise fewer than 200 nodes (e.g., 186 nodes in the Swiss system, 3657 substations for the European transmission systems https://arxiv.org/abs/1806.01613).

• A more extensive test case like the one presented in https://arxiv.org/abs/2301.08840 is synthetic and does not reflect the structure of actual grids.

Comparison with the references reported:

• Unlike the supervised methods reported, our approach does not require labeled data from traditional solvers.

• While we acknowledge the importance of scalability, this work focused on validating the framework on standard benchmarks.

6. Training times are a standard limitation of any deep learning framework. Training is a one-time process; the benefit always lies in the fast inference times once the model is trained, which is crucial for real-time applications.

7. Direct comparisons with supervised ML methodologies are not meaningful because our approach is fully unsupervised. Only one work was found in [3], which presented inequality constraint violations. Supervised approaches approximate solutions from traditional solvers, while our method directly solves the problem, finding alternative feasible solutions. [3] https://arxiv.org/abs/2210.09277

8. We respectfully disagree, as the parameter is defined in the text immediately preceding Eq. (5).

9. We acknowledge that using a more realistic data distribution could improve the model's generalizability. However, our current approach (uniform distribution) is explicitly stated in Section 4.2. This method is commonly used in literature [3],[4]. [4] "PGLearn - An Open-Source Learning Toolkit for Optimal Power Flow.

10. We agree that the phrasing could be clearer. Our claim specifically refers to inequality violations, and we will revise the future text to reflect this distinction explicitly.

11. We appreciate the final suggestions and offer the following responses:

• 10,000-Bus Systems: Testing on such large systems is a valuable future direction, but we note that real-world grids often contain fewer nodes (e.g., Swiss grid, European Transmission System).

• Realistic Data Distributions: This will be explored further in future work.

• Topology Changes: Experiments will be included in the next version accounting for topology variations.

• Training Times: While deep learning frameworks inherently require significant training times, we will explore optimizations.

• Comparisons: Direct comparisons with ML methods are challenging due to different problem formulations (unsupervised vs. supervised).

We appreciate the time and effort the reviewer has dedicated to evaluating our work. However, we find it challenging to reconcile the overall tone of the feedback, which acknowledges the paper’s soundness, presentation, and contributions, with a relatively low rating of 3. We kindly ask the reviewer to clarify this discrepancy. While we have decided not to proceed further with the submission, we would still like to take this opportunity to clarify and address the points raised.

1. The architecture employed in our work was carefully customized for the AC-OPF problem by leveraging graph transformer layers, as detailed in the Appendix. While we acknowledge that the individual components (GNNs and PINNs) are established techniques, their combination in this context is novel. Importantly, we emphasize that our contribution lies not in introducing unnecessary architectural complexity but in demonstrating that the proposed framework, built on graph-transformer-based GNNs and a PINN-based unsupervised learning approach, can achieve zero inequality violations and generalize to unseen scenarios. Moreover, to the best of our knowledge, this work is the first to successfully apply the H-PINN framework to the AC-OPF problem, a key innovation that we believe adds significant value. This type of PINN is designed for problems with hard constraints, such as the OPF, and to our knowledge, we are the first to apply this method successfully in this field.

2. We thank the reviewer for this comment. While we agree that a comparison with another IP solver (e.g., IPOPT in addition to MIPS) could be added, our work is not aimed at addressing every modeling approach to the OPF problem. Instead, we focus on demonstrating that an unsupervised ML approach can solve the full OPF formulation without relying on relaxations or simplifications.

Q1. Yes, our GNN+PINN architecture was specifically tailored for the AC-OPF problem by integrating graph transformer layers to effectively model the power grid's topology. This choice was motivated by prior work indicating the superior performance of transformer-based architectures in power systems. Furthermore, we are the first to apply the H-PINN method to solve the ACOPF problem.

Q2. While we acknowledge the importance of comparisons, methods using GNNs or PINNs in isolation are fundamentally different from our combined approach. Specifically: A GNN-only solver would require supervised learning and depend on labeled datasets generated by traditional solvers, which our method aims to eliminate. A PINN-only solver would not benefit from the structural insights provided by the GNN architecture. The strength of our framework lies in the combination of these, enabling an unsupervised solution that generalizes well. Therefore, while we agree our work could benefit from more comparison with other AC-OPF solvers, it would not be possible to compare it fairly with other ML techniques.

Q3. We believe the reviewer is requesting clarification on the necessity of using PINNs with hard constraints in the AC-OPF problem. To address this:

• Necessity of Hard Constraints: The AC-OPF problem involves both equality and inequality constraints that are critical for ensuring physical feasibility. Using a PINN with hard constraints ensures that these conditions are met during optimization, making it well-suited for this application.

• Improvement: Our method improves upon standard PINN approaches by integrating a graph-based architecture, which captures the underlying power grid topology more effectively. This, combined with the H-PINN framework, allows us to achieve zero inequality violations, a significant improvement over previous works.

Q4. We appreciate this suggestion and acknowledge the importance of prior work on physics-informed GNN-based state estimation. However, it is important to note that the focus of this paper is on AC-OPF, not state estimation. While these domains are related, they address different challenges within power systems. If the reviewer could provide specific references for the PSSE work and elaborate on why they should be discussed in a work only focused on OPF, we would be happy to review and consider including them as part of our motivation in the revised manuscript.

We thank the reviewer for their thoughtful comments and feedback. While we have decided not to proceed further with the submission, we would still like to take this opportunity to clarify and address the points raised.

1. We appreciate your feedback on this point. To clarify, we do not claim to be the first to use Physics-Informed Neural Networks (PINNs) for solving ACOPF problems. Instead, our contribution lies in successfully achieving zero inequality violation using a PINN in a fully unsupervised framework, which has not been demonstrated in the works you reference.

2. We recognize the importance of a comprehensive literature review and aim to clearly differentiate our work from existing studies. Below, we provide specific comparisons and highlight the distinctions: [1] is a review article that broadly covers end-to-end learning methods for constrained optimization. While it includes methods cited in our work and those mentioned in your comments, our focus is on unsupervised PINN approaches for ACOPF, not general constrained optimization. [2] addresses constrained optimization using a Deep Neural Network (DNN) but does not generalize to topological differences in power systems, exhibits inequality violations (unlike our approach), and is tested on only two simple systems with limited configurations (e.g., one generator per node and a single demand). [3] was reviewed in our study but not cited because the method is not fully unsupervised, a key focus of our work. [4] primarily targets general constrained optimization problems and is tested on a single power system (IEEE57) with 1000 instances, experiencing constraint violations in the test set. This limited scope does not demonstrate the robustness or generality needed for broader ACOPF applications.

3. You raise an important point about comparisons with existing ML methods. However, our method's unsupervised nature makes direct comparisons with supervised ML approaches less relevant, as those methods typically map traditional ACOPF solutions (obtained via solvers) and do not solve the problem directly. Instead, it will be more relevant to compare our method to traditional optimization solvers, including MIPS and IPOPT, as both solve the problem directly.

4. Our choice of using only 500 demand samples stems from the principle of avoiding data-intensive methodologies, a primary advantage of unsupervised learning approaches over supervised techniques. However, we acknowledge that increasing the number of samples could provide additional insights. Training with larger datasets would extend training times but does not fundamentally alter the method or its conclusions. The reference [4] reported similarly uses only 1000 instances.

5. The experiments were conducted on ETH Euler Clusters, which comprise multiple nodes. Detailed specifications can be found on the official Euler webpage. Training times vary depending on the grid size, ranging from several hours to a few days. However, it is important to note that training is a one-time computational cost, after which inference becomes significantly faster than traditional solvers. This advantage is demonstrated in Figure 4.

6. The model depicted in Figure 1 was custom-developed for this work. To the best of our knowledge, no other studies use the same architecture combined with our PINN loss function.

7. You correctly note that achieving global optimality is a limitation of nonlinear optimization problems, including ACOPF. Our method does not claim to resolve this inherent challenge. Instead, we aim to provide an alternative to traditional nonlinear solvers for ACOPF problems, demonstrating competitive performance and practical advantages, such as zero inequality violations.

8. The equality loss of up to 20 MW observed for the 118-bus system does not imply that MIPS fails to converge. Rather, it reflects the fact that equality constraints are treated as hard constraints and minor violations can occur within the solver's tolerance. Importantly, MIPS would not converge if inequality constraints were violated beyond permissible limits or if an initial condition was infeasible.

We appreciate the time and effort the reviewer has dedicated to evaluating our work. However, we find it difficult to reconcile the detailed evaluation—characterizing the paper as having fair soundness, good presentation, and a fair contribution—with a final rating of 1. We would kindly ask for clarification regarding this rating, as the provided feedback appears to reflect a more moderate assessment of the paper. While we have decided not to proceed further with the submission, we would still like to take this opportunity to clarify and address the points raised.

1. All results presented in the paper explicitly demonstrate zero inequality constraint violations, which is a key indicator of feasibility. Additionally, the equality constraint violations achieved by our method are comparable to, and often lower than, those of traditional solvers like MIPS. This is evidenced in the numerical results presented in the manuscript, particularly in Table 2. The reviewer does not provide specific arguments or examples to support this statement. We kindly ask the reviewer to elaborate further so that we can better address their concerns.

2. We believe the reviewer has misunderstood the intent of Figure 3. This figure does not show constraint violations but rather a comparison of results between our method and the traditional solver (MIPS). The ACOPF problem does not have a unique solution; different solvers may converge to different feasible solutions that represent distinct local minima. These differences often reflect trade-offs between objectives, such as operating cost and strict adherence to power flow equations. Figure 3 highlights such trade-offs. To avoid further confusion, we will revise the figure's description and include additional context in the next version of the manuscript to make its purpose clearer.

3. The values reported in Table 2 represent the maximum equality constraint violations observed across a test set of 50 grids under varying load scenarios. These values do not imply that the problem is infeasible or that operators would reject the solutions outright. For comparison, the equality constraint violations of our method (16 MW) are smaller than those of MIPS (20 MW). Additionally, the violations observed are within acceptable tolerances for large-scale nonlinear optimization problems like ACOPF. To provide additional validation, we plan to include results obtained with other solvers (e.g., IPOPT) in the next version of the manuscript.

4. As stated in the introduction, the ACOPF problem is of significant importance in power system operations and analysis due to its complexity and practical relevance. Our focus is on applying the proposed framework to this domain. While the method is general and could be extended to other optimization problems, these applications are beyond the scope of the current work.

5. You raise an important point about comparisons with existing ML methods. However, our method's unsupervised nature makes direct comparisons with most ML approaches (which are supervised) not relevant, as those methods typically map traditional ACOPF solutions (obtained via solvers) and do not solve the problem directly. Instead, it will be more relevant to compare our method to traditional optimization solvers, including MIPS and IPOPT, as both solve the problem directly.

6. Indeed, the neural network will always output a solution, even when the problem is infeasible. However, infeasibility can be detected by monitoring the violation of inequality constraints or if numerical instability during training is observed. If such violations exceed acceptable thresholds, it indicates that no feasible solution exists. In our experiments, we did not encounter cases of infeasibility, as all presented results respect inequality constraints. We will include infeasibility detection mechanisms in the future version of the manuscript.

We thank the reviewer for their thoughtful comments and feedback. While we have decided not to proceed further with the submission, we would still like to take this opportunity to clarify and address the points raised.

Novelty: We appreciate the reviewer’s feedback regarding the novelty of the work. We acknowledge that GNNs and PINNs are well-established frameworks. However, our work is novel in the following ways:

1. Application of the H-PINN framework to AC-OPF: To the best of our knowledge, we are the first to successfully apply the H-PINN framework to the AC-OPF problem and achieve zero constraint violations in an unsupervised setting. This type of PINN is designed for problems with hard constraints, such as the OPF, and to our knowledge, we are the first to apply this method successfully in this field.

2. Investigation of GNN architecture: While we utilize GNNs, we carefully evaluated different architectures, concluding that the Graph Transformer architecture is the most effective for power systems. This choice stems from prior observations and empirical results, which we highlight in our work.

Scope: We agree with the reviewer that handling topology variations is a critical consideration for real-world applications. While our current work does not explicitly address topology variations, we acknowledge this limitation and plan to address it in future work.

Specifically:

• Planned evaluation on topology changes: We aim to test our framework on datasets that include N-1 and N-k contingencies. This will help evaluate the ability of the GNN to generalize across different topologies.

• Adaptability of the GNN model: Incorporating such contingencies will allow us to explore methods to enhance the GNN's adaptability without requiring retraining for each new adjacency matrix.

ML-specific challenges in modeling multiple generators per electrical bus: The use of artificial nodes to represent multiple generators per bus is a modeling choice rather than a fundamental limitation. As noted by the reviewer, this adds complexity and requires the framework to learn a more granular solution. We observed this in prior work, where results for grids with multiple generators at the bus level were not observed.

Phase angle comparisons in the evaluation: We clarify that the absence of phase angle comparisons stems from the use of the MATPOWER data structure and the MIPS solver, which do not designate a slack node or assign a reference angle for the IEEE 118-bus system.

Context in Figure 3: The purpose of Figure 3 is to compare the predicted physical variables (e.g., voltages, powers) of the proposed framework (PINCO) with those of the MIPS solver. While generator operating cost is a central focus in OPF, respecting physical constraints such as grid properties is equally important. The figure primarily aims to illustrate how well the proposed method aligns with these physical constraints. We thank the reviewer for pointing out the need for additional context. We will revise the discussion of Figure 3 to clarify its intent and significance.

Visualizations tracking equality loss and cost differences during training: We agree with the reviewer that including plots tracking the evolution of equality loss and relative cost differences throughout training would enhance interpretability. We will include these visualizations in the revised version of the paper to provide deeper insights into the model’s alignment with the OPF objective over time.

Formulating OPF for time-varying topologies: We acknowledge the importance of addressing dynamic topologies in real-world power systems and agree with the reviewer that this represents an exciting direction for future research. While our current work focuses on fixed topologies, we plan to extend the framework to accommodate time-varying grid topologies. This will involve evaluating the method on datasets that incorporate real-world operational scenarios, including topology changes and contingencies.