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Expediting AC Contingency Analysis using a Basecase Machine Learning Model

Md Obaidur Rahman, Vassilis Kekatos

arXiv:2609.36510v1eess.SYcs.SY

TL;DR

AC contingency analysis is computationally demanding, and existing ML approaches can require training across scenarios and topologies. This paper uses fixed-point iterations to reuse one basecase-trained ML model for post-contingency states. Tests on a 6,717-bus synthetic Texas system report convergence across all non-critical line outages, with prediction errors close to basecase.

  • Problem

    AC contingency analysis is computationally demanding, and training ML models across multiple scenarios and topologies remains a major burden.

    Source: §I · pp. 1–2

  • Method

    The paper uses a fixed-point framework that reuses a single ML model trained solely on basecase topology data to predict post-contingency AC-PF states.

    Source: §I · p. 2

  • Results

    On the 6,717-bus Texas7k system, the iteration converges for all non-critical line contingencies, with prediction errors remaining close to basecase.

    Source: §I · p. 2, §VII · p. 8

  • Takeaways & Limitations

    The method offers a favorable speed–accuracy tradeoff, reducing runtime relative to NR while maintaining higher accuracy than the DC-based solver.

    Source: §I · p. 2

    The demonstrated convergence scope is non-critical single-line outages.

    Source: §VII · p. 8

Abstract

from arXiv · show

Grid planning and operation under increasingly variable operating conditions require fast and accurate AC power flow (AC-PF) contingency analysis for numerous line outages. While ML models can accelerate these computations, existing approaches often require either training separate models for different contingencies or a single model using data from multiple topologies. Both approaches incur substantial offline data generation and training costs. To address this gap, this work proposes a fixed-point framework that reuses a single ML model trained exclusively on basecase topology data to predict post-contingency AC-PF states under any non-critical single-line outage. We analyze the proposed method's convergence under the DC model approximation and derive its convergence rate in terms of network parameters. As a side result, we show that power transfer distribution factors (PTDFs) for non-critical lines have magnitudes less than one. We further characterize the ML training region to accommodate PF specifications encountered during fixed-point iterations and derive bounds on ML prediction errors for post-contingency state estimates. Numerical tests on a 6,717-bus synthetic Texas system demonstrate convergence across all non-critical single-line outages, with prediction errors remaining close to those of basecase. The proposed framework offers a favorable tradeoff between the speed of DC solvers and Newton-Raphson accuracy.

I. INTRODUCTION

AC contingency analysis is computationally demanding, while existing data-driven approaches can require training across many scenarios or topologies. This work reuses a basecase-trained ML model through fixed-point iterations to estimate post-contingency AC-PF states, with convergence and accuracy evaluated on a large synthetic system.

I. INTRODUCTION

  • Prior approaches include DC-based screening, AC linearizations, and learned contingency models, while training across multiple scenarios and topologies remains computationally burdensome.Some AC methods depend on the linearization point or reference topology, and a cited contingency-screening model requires training on numerous scenarios and topologies.Source: §I · pp. 1–2
  • A single ML model trained exclusively on basecase topology data is reused in a fixed-point iteration to predict post-contingency AC-PF states.This addresses the research question of whether a basecase-trained model can be reused for post-contingency predictions.Source: §I · p. 2
  • Under the DC-PF approximation, the work establishes convergence for all non-critical single-line outages and derives bounds on PTDFs.The introduction identifies bounding PTDFs as a byproduct of the convergence analysis.Source: §I · p. 2
  • The work characterizes expanded AC-PF specifications for basecase-model training and analyzes how ML prediction errors propagate to post-contingency state estimates.These contributions address the specifications encountered during fixed-point iterations and the resulting state-estimation errors.Source: §I · p. 2
  • On the Texas 6,717-bus system, the method converges for all non-critical line contingencies and substantially reduces runtime relative to NR while achieving higher accuracy than the DC-based solver.The reported prediction errors remain close to those of basecase.Source: §I · p. 2
  • Compared with the conference precursor’s SDP-based analysis, the DC-PF analysis applies to arbitrary ML models, does not depend on the pretrained model, and can indicate which outages may need more iterations or training data.The precursor’s SDP analysis was computationally demanding, limited to DNNs, and could yield loose sensitivity bounds.Source: §I · p. 2

II. PROBLEM STATEMENT AND FORMULATION

The section formulates post-contingency AC-PF states through the basecase mapping, then uses that relation to motivate a fixed-point approach with one basecase-trained ML model. The approach targets non-critical single-line outages and applies to preventive and corrective analysis.

II. PROBLEM STATEMENT AND FORMULATION

  • Contingency analysis requires solving L + 1 AC-PF problems, and the proposed formulation applies to both preventive and corrective settings.Preventive analysis uses the same specification vector across contingencies, while corrective analysis may vary specifications by contingency.Source: §II · p. 2
  • The fixed-point relation recasts the contingent AC-PF equations as basecase equations with injections modified by a state-dependent compensation term.The compensation term is nonzero at the outaged line’s terminal buses; the section contrasts this quadratic AC-PF formulation with the linear DC-PF compensation trick.Source: §II · p. 3
  • Each fixed-point step modifies 2–4 specification entries using the previous estimate, then evaluates the basecase mapping to produce the next state estimate.The corresponding AC-PF iteration offers no practical advantage over solving the contingent case directly; its value is motivating the ML version.Source: §II · p. 3
  • One ML model trained on basecase topology can estimate states for any contingency by repeatedly applying the basecase model with contingency-specific feedback.For a single-line outage, the feedback affects only the outaged line’s terminal buses.Source: §II · p. 3, Figure 1 · p. 3
  • The method avoids training L additional models, while inference is attractive when iterations converge quickly and ML inference is faster than an AC-PF solver.The fixed-point loop’s feedback depends on the particular contingency.Source: §II · p. 3, Figure 1 · p. 3

III. CONVERGENCE RATE ANALYSIS

The section analyzes convergence of the ML fixed-point iteration through a DC-PF approximation. It uses this tractable analysis to identify factors governing convergence in the original AC and ML iterations.

III. CONVERGENCE RATE ANALYSIS

  • Directly proving convergence is challenging because the iteration couples an ML model with the nonlinear contingency mapping.A contraction argument would require characterizing the smoothness of the ML mapping through its Lipschitz constant.Source: §III · p. 3
  • Existing convergence results for the AC-PF solver apply only to restricted settings, including lossless or radial networks and limited loading conditions.Bounding the ML mapping’s sensitivity with an SDP can be computationally demanding, DNN-specific, and overly conservative.Source: §III · p. 4
  • The DC-PF approximation decouples the basecase mapping into P−θ and Q−V components, exposing factors that govern convergence of the AC and ML iterations.The analysis approximates the AC-PF mapping with the linearized DC-PF mappings.Source: §III · p. 4

A. Convergence of the P−θ Mapping

The P−θ fixed-point iteration converges geometrically for every non-critical line, with its convergence rate determined by the line’s self-PTDF. Lines whose self-PTDF is near one may require more iterations.

A. Convergence of the P−θ Mapping

  • The section supplies a formal proof for a widely used PTDF property and applies it to establish the self-PTDF result needed for convergence.The theorem’s relevant PTDF triplet uses the outaged line and its terminal buses.Source: A. Convergence of the P−θ Mapping · p. 4
  • For any non-critical line ℓ, the P−θ fixed-point iteration (19) converges geometrically to the solution of (18), and the self-PTDF βℓ lies in (0, 1).The initial-point condition is described as mild.Source: A. Convergence of the P−θ Mapping · p. 4
  • Lines with self-PTDF values near one may require more iterations for the P−θ fixed-point iteration to converge.The self-PTDF is PTDF_ℓ,m,n, where m and n are the terminal buses of line ℓ.Source: A. Convergence of the P−θ Mapping · p. 4

B. Convergence of the Q−V Mapping

The Q−V fixed-point iteration converges for all non-critical line outages, with its eigenvalue bounded below one. The section also reports that the contraction constants are below one across the non-critical lines of the Texas7k system.

B. Convergence of the Q−V Mapping

  • The Q−V analysis treats the reference bus as PV because its voltage magnitude is specified and its active-power injection is immaterial to this model.The iteration solves for unknown voltage magnitudes at PQ buses.Source: B. Convergence of the Q−V Mapping · p. 5
  • The Q−V fixed-point iteration in (28) converges to the solution of (27) for any non-critical line ℓ=(m,n)∈L0, with eigenvalue 0 ≤ λℓ < 1.Once uP is known, qV can be found as qV = BVPuP + BVVuV.Source: B. Convergence of the Q−V Mapping · p. 5
  • For the 6,717-bus Texas7k system, contraction constants β_ℓ and λ_ℓ are below one across all 8,079 non-critical lines.The system has 8,079 non-critical lines out of 9,140 transmission lines.Source: B. Convergence of the Q−V Mapping · p. 5

IV. ADMISSIBLE REGION FOR ML MODEL INPUTS

The ML model must cover an expanded training region so its predictions remain accurate for the modified specifications encountered during fixed-point iterations. The section gives a tractable outer approximation for this region when the active- and reactive-power specification sets are polytopes.

IV. ADMISSIBLE REGION FOR ML MODEL INPUTS

  • The model’s training region must include specification vectors generated during fixed-point iterations, not just the operator’s original study region.The method seeks an expanded region C′ containing C and the modified specifications encountered across contingencies and iterations.Source: §IV · p. 5
  • The expanded region accommodates the specification changes induced by contingencies, allowing one model to be trained for fixed-point analysis over the original study region.The construction uses polytopic sets for active and reactive power specifications.Source: §IV · pp. 5–6
  • Under the DC-PF approximation, decoupled P−θ and Q−V models let the expanded training set retain a Cartesian-product structure.The specification sets comprise active power, reactive power, and voltage magnitudes.Source: §IV · p. 5
  • For polytopic active- and reactive-power specifications, the method constructs outer approximations by solving (Ip + Iq)L linear programs.The approximation may be conservative; the reported tests nevertheless train an ML model with O(N^2) parameters accurately over the expanded region.Source: §IV · p. 6

V. PREDICTION ACCURACY OF FIXED-POINT SOLUTION

This section bounds the difference between AC-PF-based and ML-based fixed-point trajectories. Under contraction and approximation assumptions, the ML iteration remains near the post-contingency solution, with the bound depending on prediction error and a contingency-specific Lipschitz constant.

V. PREDICTION ACCURACY OF FIXED-POINT SOLUTION

  • The analysis bounds the gap between ML-based and AC-PF-based fixed-point trajectories, assuming they start from the same voltage state.It presumes the ML model approximates the basecase AC-PF mapping over the expanded specification region.Source: §V · pp. 6–7
  • The bound applies when the ML mapping is contractive in voltage, with Lipschitz constant Lℓ < 1, and approximates the basecase AC-PF mapping within error ϵ.The Lipschitz constant is defined over a voltage region where modified specifications remain in C′.Source: §V · p. 6
  • Although the trajectory gap can grow with iteration count, it remains bounded; if the AC-PF iteration converges, the ML iteration converges within a neighborhood of the post-contingency state.The neighborhood radius depends on prediction accuracy ϵ and the contingency-specific Lipschitz constant, and may be overestimated when that constant is conservative.Source: §V · p. 7

VI. NUMERICAL TESTS

On the Texas7k tests, the DNN fixed-point method offers a speed–accuracy tradeoff between the DC solver and Newton–Raphson. It converges across tested contingencies and retains prediction errors comparable to the basecase.

VI. NUMERICAL TESTS

  • From one year of Texas7k 5-minute operating scenarios, k-medoids selected 2,000 representatives; 200 defined specification set C, while 120,000 generated scenarios from ˆC were combined with the remaining 1,800 basecase scenarios to train the DNN.Source: §VI · p. 7
  • All tested contingencies converged within 20 iterations using the DNN fixed-point iteration with Anderson acceleration.The iteration was initialized at the basecase DNN solution, and contingencies were ranked by iteration count averaged across scenarios.Source: §VI · p. 7, Figure 3 · p. 7
  • The fitted empirical convergence-rate trend decreases as the P−θ spectral radius βℓ increases, in agreement with Theorem 2.Rates were estimated from the negative slope of fitted convergence curves and plotted against the corresponding spectral radii.Source: §VI · p. 7, Figure 4 · p. 7
  • Post-contingency voltage MSE remains below 0.3% and AC-PF-specification NMAE below 6% across contingencies, comparable to the basecase.The figure compares the basecase with 29 contingencies across tested scenarios.Source: §VI · p. 7, Figure 5 · p. 8
  • The proposed method is substantially faster than NR and has lower voltage MSE than the DC-based solver, which remains fastest, yielding a favorable speed–accuracy tradeoff.Runtime and voltage-error distributions were compared across testing scenarios and non-critical contingencies.Source: §VI · p. 8

VII. CONCLUSIONS AND FUTURE WORK

The paper concludes that one ML model trained on basecase data can support fixed-point AC contingency analysis, with convergence and prediction-error guarantees developed under the DC-PF approximation. Texas7k tests show convergence across non-critical line outages and errors close to basecase.

VII. CONCLUSIONS AND FUTURE WORK

  • The proposed formulation reuses a single ML model trained solely on basecase data and establishes convergence guarantees and convergence-rate characterization under the DC-PF approximation.It also characterizes an expanded training region and bounds prediction errors for post-contingency estimates.Source: §VII · p. 8
  • Texas7k tests demonstrate convergence across all non-critical line outages, with prediction errors remaining close to basecase.The paper identifies extending the formulation to N−k contingencies as future work.Source: §VII · p. 8
  • The proposed method offers a favorable tradeoff between the speed of the DC model and NR accuracy.The comparison is reported for the Texas7k system.Source: Figure 6 · p. 8, §VII · p. 8

APPENDIX

The appendix establishes bounds on PTDFs for non-critical lines and fixed-point convergence under stated conditions. It also shows that a PV-PV line outage reaches the solution after one update.

  • For a unit transfer from source i to sink j, the proof identifies the corresponding flow on line ℓ with PTDFℓ,i,j.Source: APPENDIX · p. 8
  • |PTDFℓ,i,j| < 1 for non-critical lines; a line alone crossing the cut has |PTDFℓ,i,j| = 1 and is critical.The cut argument bounds the line flow for a unit transfer; equality identifies the critical-line case.Source: APPENDIX · p. 9
  • When βℓ < 1, the fixed-point iteration converges to θℓ, provided ˜Gℓ(θ0ℓ−θℓ) ≠ 0.Source: APPENDIX · p. 9
  • With both terminal buses PV, the fixed-point iteration reaches the solution after a single update.Source: APPENDIX · p. 9

VV ˜BVP

For every non-critical line outage, the proof establishes a spectral radius below one and characterizes the fixed-point iterates as convex combinations that inform the required training region.

VV ˜BVP

  • Across all three terminal-bus cases, the spectral radius of G_ℓ satisfies 0 ≤ λ_ℓ < 1 for any non-critical line.The proof also uses that B_PP is positive semidefinite to establish λ_ℓ ≥ 0.Source: VV ˜BVP · p. 9
  • Equation (49) makes each iterate a convex combination of the initial specification and its transformed value, so C′_p should include the segment between them for every ℓ.Source: VV ˜BVP · p. 10
  • The trajectory comparison proof applies the triangle inequality and bounds its second term by ε when the relevant specification lies in C′.The iterations are initialized at the same voltage point.Source: VV ˜BVP · p. 10
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