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Scalable Self-Supervised Learning for Multiphase AC-OPF in Distribution Systems with Topology Reconfiguration

Hoang T. Nguyen, Shaohui Liu, Reetam Sen Biswas, Varsha Pendyala, Nurali Virani, Deepjyoti Deka, Priya L. Donti

arXiv:2608.25095v1eess.SYcs.LGmath.OC

TL;DR

Existing learning-based AC-OPF surrogates do not adequately address the multiphase, unbalanced, reconfigurable, and large-scale nature of distribution feeders. Penalty+SLFS provides a self-supervised surrogate with differentiable power flow and inference-time feasibility repair, achieving high-quality solutions and up to three orders of magnitude speedups over IPOPT across feeders from 13 to 8,500 nodes.

  • Problem

    Learning-based AC-OPF surrogates have not translated transmission-system speedups to multiphase, unbalanced, switch-reconfigurable distribution feeders at utility scale.

  • Method

    Penalty+SLFS trains a neural surrogate directly from AC-OPF objectives and constraints through differentiable fixed-point power flow, then applies sequential linearized feasibility seeking at inference.

  • Results

    Across feeders from 13 to 8,500 nodes, Penalty+SLFS achieves high-quality feasible solutions and up to two to three orders of magnitude inference speedups over IPOPT.

  • Takeaways & Limitations

    The results support surrogate learning as a potential approach for fast, feasible, and robust approximations to multiphase distribution AC-OPF.

Abstract

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The proliferation of distributed energy resources (DERs) in distribution grids enables the active coordination of these assets to reduce costs and enable cleaner operations. Realizing this potential requires solving multiphase AC optimal power flow (AC-OPF) quickly across varying loads, DER availabilities, and topology reconfigurations, at much greater speed and scale than conventional nonlinear solvers. Learning-based surrogates can offer millisecond inference, yet existing methods target largely balanced transmission systems and do not scale to the multiphase, unbalanced, and reconfigurable nature of distribution feeders at utility scale. We present the Penalty + Sequential Linearized Feasibility Seeking (SLFS) algorithm, a self-supervised learning framework for multiphase distribution AC-OPF under switch-induced topology changes. Penalty+SLFS requires no labeled optimal solutions and trains directly from the AC-OPF objective and constraints through a differentiable fixed-point power flow solver, avoiding expensive label generation and admitting robust training procedures. Topology changes are handled efficiently using Sherman-Morrison-Woodbury updates of the admittance-matrix inverse, while an M-step Jacobian approximation accelerates differentiation through the power flow solver. At inference, SLFS repairs any infeasible predictions, providing feasibility guarantees with low computational overhead. On IEEE feeders ranging from 13 to 8,500 nodes, Penalty+SLFS achieves negligible optimality gaps and near-zero constraint violations, delivers up to three orders of magnitude speedups over IPOPT, and remains robust under large distributional shifts, demonstrating a viable path toward real-time, topology-aware AC-OPF for large-scale distribution grids.

I. INTRODUCTION

Distribution AC-OPF surrogates must address multiphase unbalanced physics, switch-induced topology changes, and large feeders while maintaining feasibility. Penalty+SLFS is introduced as a self-supervised approach targeting these requirements without labeled optimal solutions.

  • Motivation: Distribution feeders require near-real-time evaluation across varying loads, DER availabilities, and switch configurations under multiphase, unbalanced operating constraints.These conditions tighten voltage and current limits and alter power flows through topology changes.
  • Motivation: Direct solution mappings can deliver millisecond inference and two to three orders of magnitude speedups in transmission AC-OPF, but these gains have not translated to distribution feeders.The gap motivates methods designed for distribution-system scale and complexity.
  • Feasibility challenge: Penalty losses improve constraint reliability but do not strictly enforce feasibility, whereas hard enforcement can be substantially more expensive during training and inference.This creates a trade-off between computational cost and guaranteed feasibility.
  • Contributions: Penalty+SLFS trains a self-supervised surrogate directly with a differentiable power-flow solver, penalty-based loss, and inference-time feasibility seeking.The approach requires input samples rather than labeled optimal OPF solutions.
  • Validation scope: The framework targets multiphase distribution systems with topology reconfiguration and scales across grids ranging from 13 to 8,500 nodes.The introduction reports inference speedups of up to two to three orders of magnitude over IPOPT on medium-to-large feeders.

III. PROPOSED METHOD

Penalty+SLFS maps operating conditions to clipped DER setpoints, evaluates them through differentiable fixed-point power flow, and repairs residual network violations with SLFS. Training efficiency comes from truncated M-step differentiation, while topology-dependent admittance inverses are updated efficiently.

  • Pipeline: The neural network takes loads, DER availability, switch configuration, and objective weights as inputs, then outputs raw DER active and reactive setpoints.A clip layer converts raw outputs into setpoints satisfying DER operational limits.
  • Feasibility seeking: At inference, SLFS linearizes network constraints around the current operating point and solves damped proximal feasibility subproblems to repair violations exceeding tolerance ϵ.The procedure iteratively refines DER setpoints rather than changing the neural-network prediction directly.
  • Power-flow solver: The differentiable fixed-point power-flow solver computes voltages used to evaluate the AC-OPF objective and remaining substation, voltage, and line constraints.Iterations continue until successive voltage estimates fall below a predefined threshold.
  • Differentiation: The M-step Jacobian approximation drops earlier voltage sensitivity terms, reducing differentiation cost while its approximation error decays exponentially with M under the stated assumptions.The analysis relies on contraction and local Lipschitz conditions for the fixed-point mapping.

D. Admittance Inverse via SMW

The method accelerates topology-dependent admittance-inverse updates by representing switch changes as low-rank perturbations around a reference configuration. Iterative refinement addresses numerical error when these updates are ill-conditioned.

  • Reference topology: A reference switch configuration z0 is precomputed, and only differing switch contributions are added or removed for each topology sample.This avoids rebuilding the admittance matrix from scratch.
  • Low-rank switch updates: Each closed switch contributes a sparse branch stamp supported only on its incident phase-node rows and columns, with rank at most 2|Φmn|.The low-rank structure enables the Woodbury factorization.
  • Low-rank switch updates: Switch-status changes are stacked into U(z) and V(z), whose update dimension nupd is typically much smaller than the number of phase nodes |P|.Only a few switches usually change per topology sample.
  • Computational cost: Each topology update costs O(nupd^3 + |P|^2nupd), compared with O(|P|^3) for dense inversion when nupd ≪ |P|.The method explicitly inverts only the small correction matrix and reuses the precomputed reference inverse.
  • Numerical refinement: Roundoff can be amplified when K(z) is ill-conditioned, so one or two SMW iterative-refinement steps correct deviations without forming a dense inverse.The issue is strongest for high-impedance tie switches or near-parallel paths.

E. Loss Function

The loss evaluates cost and network-constraint violations after clipped DER predictions pass through a differentiable fixed-point power-flow solver. A fixed penalty weight balances cost against feasibility, while infeasible predictions can be repaired at inference.

  • Loss construction: The penalty loss combines the AC-OPF objective with positive-part penalties for violations of stacked network operational constraints.The constraints include voltage, branch-current, angle-difference, and substation limits.
  • Loss construction: The clip layer enforces DER box, availability, and apparent-power limits, while the forward power-flow solver imposes the power-flow equalities.The remaining network inequalities are evaluated at the resulting operating point.
  • Penalty weighting: A fixed ρ is used because Lagrangian updates can grow monotonically on minibatches containing infeasible instances and overemphasize feasibility relative to cost.Keeping ρ fixed also simplifies hyperparameter selection.
  • Inference repair: If residual violations exceed tolerance ϵ at inference, SLFS repeatedly linearizes the network constraints and solves damped proximal feasibility subproblems.This repair stage follows the network prediction and is separate from the training loss.
  • Differentiable evaluation: All linearization matrices are formed at the current reference operating point and recomputed whenever SLFS advances to a new iterate.The reference includes the current DER setpoints, voltages, and injections.

2) Algorithm and Convergence Analysis:

SLFS repairs predicted DER setpoints by repeatedly linearizing nonlinear constraints and solving convex, damped proximal feasibility subproblems. Its convergence analysis shows monotonic violation reduction when the proximal step moves the iterate.

  • SLFS procedure: SLFS iteratively linearizes nonlinear constraints around the current operating point and solves a convex feasibility subproblem for a fixed number of iterations.The procedure is designed to reduce constraint violations rather than optimize a new objective.
  • SLFS procedure: Algorithm 2 starts from predicted DER setpoints, solves power flow, constructs the local linearization, applies a proximal step, damps the update, and repeats.The final iterate yields approximately feasible active- and reactive-power setpoints.
  • Feasibility enforcement: The feasible set F contains DER box, availability, and apparent-power constraints, and CLIP maps any point into F by construction.CLIP is reused inside the primal-dual feasibility solver.
  • Proximal subproblem: The proximal term makes the convex feasibility objective strongly convex with modulus η, giving the subproblem a unique solution.The regularizer controls the step around the current iterate.
  • Numerical implementation: PDHG solves each proximal subproblem using matrix-vector products with A(xk) and A(xk)⊤, elementwise clipping, and the CLIP feasibility map.This avoids linear solves and explicit Jacobian materialization.
  • Convergence: When the proximal step moves the iterate, the violation decreases monotonically under the stated linear-model assumption and damping choice.The convergence result is conditioned on Assumption 2 and the prescribed damping factor.

IV. EXPERIMENTAL RESULTS

Experiments evaluate Penalty+SLFS and related methods on IEEE feeders from 13 buses to 8,500 nodes, using solution quality, speed, extreme scenarios, and robustness tests. Penalty+SLFS provides strong feasibility and runtime performance across these evaluations.

  • Experiment Setup: 13-bus to 8,500-node IEEE feeders are evaluated with 200,000 training samples and 2,000 samples per test set.The feeders contain varying numbers of PV units and switches, while test scenarios include base, extreme, and out-of-distribution conditions.
  • Solution quality and speedup: Direct Penalty fails to converge on every feeder, while the remaining learning-based methods succeed on the base test set.Its severe infeasibility makes negative relative optimality gaps uninterpretable as improved solutions.
  • Solution quality and speedup: Penalty+SLFS reduces constraint violations by one to three orders of magnitude versus Penalty, with only a modest increase in absolute cost error.It also outperforms Linearized OPF on every feeder, with mean violations at least 7× and up to three orders of magnitude lower and cost errors and gaps 1.6–7.4× smaller.
  • Solution quality and speedup: On infeasible instances, Penalty+SLFS and SLFS-FSNet attain constraint violations comparable to IPOPT.Optimality gaps should be interpreted cautiously because infeasible solvers may return different points with similar violation levels.
  • Solution quality and speedup: 780× and 1121× speedups over IPOPT are achieved by Penalty+SLFS on the two largest feeders.On feasible test sets, it is faster than SLFS-FSNet because repair can be skipped when predictions are already feasible.
  • Solution quality and speedup: Penalty+SLFS maintains the strongest overall feasibility and runtime profile, supporting learning-based surrogates as an alternative to Linearized OPF for this problem class.The reported summary covers improved feasibility over Penalty, typical speed advantages over SLFS-FSNet, and up to three orders of magnitude speedup over IPOPT.

2) Robustness to extreme regimes and distributional shift:

Penalty+SLFS maintains low violations and small optimality gaps across extreme operating regimes, distributional shifts, feeder sizes, and training distributions. Its scalable differentiation, matrix updates, and feasibility seeking support efficient training and inference.

  • Extreme regimes: 0.25% is the largest optimality gap in the high-load/low-DER regime on the 8500-node feeder.Violations are lowest for Penalty+SLFS, while cost errors and gaps remain comparable to Penalty and SLFS-FSNet.
  • Distributional shift: Below 1% optimality gaps persist at all tested distribution-shift levels, while Penalty+SLFS constraint violations remain largely flat.The test set contains 2,000 samples per feeder at each Gaussian noise level, with noise mean equal to standard deviation.
  • Computational scaling: 2.74× faster differentiation is achieved by the 3-step Jacobian approximation on the 906-bus feeder while retaining sufficient training accuracy.The approximation is adopted as a practical balance between accuracy, runtime, and memory.
  • Computational scaling: 28× SMW speedup is reached on the 8500-node feeder compared with direct admittance-matrix inversion at batch size 8.SMW speedups increase with feeder size, while direct inversion reaches 1.7 s on the largest feeder.
  • Training distributions: Training-mix changes have little effect on violations, which are typically 10^-4–10^-5 on the base test set, while optimality gaps vary by regime and feeder.On the 240-bus feeder, the low-load/high-DER gap reaches about 1.4% at W2; broader training ranges are recommended when data generation is inexpensive.

APPENDIX

Appendix experiments examine penalty weighting, topology inputs, neural-network backbones, and batch-size scaling. They show that topology information improves optimality, MLPs outperform GCNs on gaps, and batching improves throughput most on smaller feeders.

  • Penalty weight: ρ = 10^-5 yields the smallest absolute cost and optimality gaps, while violations decrease as the penalty weight increases.The most aggressive tested setting, ρ = 10^6, typically has the largest cost and optimality gaps.
  • Topology inputs: Switch-status inputs substantially reduce absolute cost and optimality gaps while preserving feasibility.The ablation underscores the value of topology information in the neural-network input.
  • Backbones: MLP and GCN backbones achieve similar feasibility, but MLP consistently attains lower absolute cost and optimality gaps.Both backbones use the same training pipeline and budget, while SLFS enforces constraints independently of the backbone.
  • Batch scaling: 4.5–9.7× wall-time growth from batch size 1 to 64 produces 6.6–14× higher per-sample throughput on 13-, 123-, and 240-bus systems.Throughput reaches up to 1.6 × 10^4 solutions/s on the 13-bus feeder; larger feeders scale linearly to slightly superlinearly.

C. SLFS-FSNet

SLFS-FSNet embeds sequential linearized feasibility seeking in both training and inference to transform raw DER predictions into feasible setpoints. SMW updates extend the fixed-topology FSNet design to switch-induced topology changes.

  • SLFS-FSNet: SLFS-FSNet maps raw DER setpoints p_der,raw and q_der,raw to feasible setpoints p_der and q_der.The SLFS module is embedded during both training and inference to enforce constraint satisfaction.
  • Topology adaptation: SMW updates and SLFS replace FSNet’s fixed-topology Levenberg–Marquardt feasibility solver for handling topology changes and accelerating feasibility seeking.The original FSNet assumes a fixed topology, whereas the adapted design accounts for reconfiguration.

D. Linearized network constraints

The appendix constructs linearized network constraints for voltage, angle difference, branch current, and substation power limits. These blocks are assembled into a full affine constraint model used by the feasibility-seeking pipeline.

  • Angle-difference constraints: The affine voltage model is extended to all phase nodes so branch angle-difference constraints can be linearized.The construction uses ξ_mn := v_m v_n and κ_mn := tan(α_max,mn), then stacks the resulting terms across branches.
  • Substation-power constraints: Because slack voltage v_0 is fixed, substation injection s_0 depends on x only through load-bus voltages v.The substation-power block applies the same magnitude linearization used for other network constraints.
  • Branch-current constraints: Branch-current limits are linearized from i_mn = y_mn(v_m − v_n) using the extended affine voltage model and magnitude linearization.The resulting branch-current blocks are represented through stacked reference currents and Jacobian terms.
  • Full linearized model: Voltage-magnitude, substation-power, angle-difference, and branch-current blocks are stacked into A(bx)x + b(bx).The full model combines the voltage-magnitude Jacobian A_v(bx) with the additional network-constraint blocks.

E. Proof of Lemma 1 and Proposition 1

The proofs establish norm bounds and Lipschitz properties for the power-flow mapping, then use auxiliary matrix inequalities and Jacobian approximations to bound the fixed-point sensitivity error.

  • Proof of Lemma 1: Lemma 2 provides a complex-valued reciprocal-difference inequality under lower bounds on the magnitudes of a and b.The proof applies the inequality componentwise with |(v1)i| and |(v2)i| bounded below by vmin.
  • Proof of Lemma 1: The proof of Lemma 1 bounds the s-Jacobian, v-Jacobian, and Lipschitz continuity of the v-Jacobian using inverse-admittance expressions and induced norms.Perturbations in s and v yield explicit differential expressions for G, from which supremum norm bounds are obtained.
  • Proof of Lemma 1: The resulting Jacobian difference is bounded by 2(vmin)^-3∥v1−v2∥∞ after applying submultiplicativity and the componentwise lower bounds.This establishes the required Lipschitz estimate for the v-Jacobian.
  • Proof of Proposition 1: Lemma 3 bounds the difference between resolvent products using ∥A∥∞<1, while Lemma 4 applies chain-rule bounds to the fixed-point Jacobians.Lemma 4 introduces A⋆=JvG(v⋆(s),s) and B⋆=JsG(v⋆(s),s) for the sensitivity analysis.

F. Proof of Theorem 1

The proof of Theorem 1 builds on Lipschitz penalty properties and Assumption 2 to derive descent-related bounds for the sequential linearized update.

  • Proof of Theorem 1: Lemma 6 shows that ϕ(y)=[y−u]+ + [l−y]+ is 1-Lipschitz continuous for vector arguments.The scalar derivative or subgradient is bounded by 1, and the vector result follows accordingly.
  • Proof of Theorem 1: Under Assumption 2, Lemma 7 derives a bound using the triangle inequality and the Lipschitz property of ϕ.The proof explicitly invokes Assumption 2 after applying Lemma 6.
  • Proof of Theorem 1: Lemma 8 bounds the update rule through a quadratic term in ∆xk and a higher-order term involving ∥∆xk∥∞^(1+ν).The displayed bound contains η/2∥∆xk∥2^2 and (αk)^(1+ν)δ∥∆xk∥∞^(1+ν).
  • Proof of Theorem 1: The theorem proof uses optimality of the linearized update, consistency at xk, convexity of Vlin, and Lemma 7 to establish the required inequality.The chain of inequalities invokes Vlin(xk;xk)=V(xk) and the preceding update bounds.
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