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Certified Safety Radii in Forecast-Error Space for Wasserstein Distributionally Robust Small Signal Stability-Constrained AC Optimal Power Flow via Lifted Spectrahedral Containment

Ziqi Zhang, Xi Chen

arXiv:2608.30201v1cs.LG

TL;DR

The paper targets the difficulty of obtaining decision-dependent sample-to-failure distances for small-signal-stability-constrained AC-OPF under Wasserstein ambiguity. It certifies uncertainty balls through lifted PSD containment and nonlinear PF enclosures, then couples the resulting radii to Wasserstein risk reformulation. Numerical studies report close tracking of first stability failure with a strict inner safety margin.

  • Problem

    Decision-dependent AC equilibria make distances from forecast-error samples to combined PF, operating-limit, and small-signal-stability failure implicit and difficult to embed in Wasserstein-robust OPF.

    Source: §I · p. 1, B. Research Gap and Contributions · p. 2

  • Method

    The framework combines component-Perron PF tubes, exact adjoint affine-quadratic representations, matrix remainder bounds, and lifted PSD containment to certify sample-wise safe radii in forecast-error space.

    Source: §I · p. 1, B. Research Gap and Contributions · p. 2, §III · p. 4

  • Results

    The certified radius is 0.03993374, capturing 99.8344% of the exact distance with 0.1656% relative conservatism; fitted growth rates agree with spectral abscissae within 4.21 × 10−6.

    Source: A. Mechanism Study: Certified Radius… · p. 8

  • Takeaways & Limitations

    Certified radii provide rigorous lower bounds usable directly in the distance-based Wasserstein risk model without explicitly constructing the nonlinear failure boundary.

    Source: §VI · p. 9

Abstract

from arXiv · show

Directly robustifying small-signal stability in AC optimal power flow is challenging since the stability boundary in the original uncertainty space is implicit, highly nonconvex, and changes with the operating decision. This paper exploits an alternative geometry. For a fixed model-specific stability certificate admitting suitable physical lifts, the small-signal stability requirement becomes an affine positive semidefinite constraint in the lifted variables, thereby defining a convex certified safe region. Instead of approximating the nonlinear instability boundary itself, we optimize a sample-wise safe radius in the original uncertainty space and certify, in the lifted space, that the entire power-flow image of the corresponding uncertainty ball is contained in the convex stability region. To this end, a componentwise Perron certificate guarantees existence, uniqueness, and Jacobian regularity of the target AC power-flow branch throughout each ball. An adjoint elimination then provides an exact affine-quadratic representation of the stability-relevant quantities, while rigorous matrix remainder bounds convert their nonlinear variation into finite robust PSD constraints. The resulting radii are certified lower bounds on the distances from empirical samples to failure and can therefore be coupled directly to the distance-based reformulation of a Wasserstein distributionally robust chance constraint, without directly approximating the instability boundary. Numerical studies demonstrate the effectiveness of the proposed framework.

I. INTRODUCTION

The paper addresses the missing interface between forecast-error Wasserstein distances and decision-dependent AC power-flow stability by certifying sample-wise safe radii through lifted convex geometry.

Research gap

  • Wasserstein chance constraints require sample distances to failure in forecast-error coordinates, but AC-OPF stability failure is hidden behind a decision-dependent nonlinear equilibrium map.Failure may involve branch loss, PF-Jacobian singularity, operating-limit violation, or loss of small-signal stability.Source: §I · p. 1, B. Research Gap and Contributions · p. 2

Core approach

  • The method certifies each weighted-ℓ1 uncertainty ball’s complete nonlinear PF image inside a convex lifted stability region instead of approximating the full failure boundary.The statistical data and transportation metric remain in the original forecast-error space.Source: §I · p. 1

Contribution

  • Certified radii rigorously lower-bound sample-to-failure distances for PF-branch, regularity, operating-limit, and selected stability-certificate failures.These radii provide the sample-wise distance interface needed by the Wasserstein risk model.Source: B. Research Gap and Contributions · p. 2

Certification machinery

  • A component-Perron PF tube, exact adjoint elimination, and matrix-level quadratic-remainder certificates reduce whole-ball safety to finite SDP/SOCP constraints.The construction preserves safety-output cancellations while certifying robust PSD containment.Source: B. Research Gap and Contributions · p. 2

Optimization framework

  • The end-to-end WDRO-OPF jointly optimizes dispatch and certified radii in a sequential conic master, avoiding embedded closest-failure computations.Fixed local bounds yield an SDP/SOCP, but sequential updates do not provide a global optimum of the original nonlinear problem.Source: B. Research Gap and Contributions · p. 2

Lifted certificate

  • The fixed model-specific stability certificate becomes an affine PSD condition in lifted equilibrium quantities, defining a convex spectrahedral feasible set.The buffered condition Kcase(z, u) ⪰τKI is used as the inner certificate.Source: A. Fixed Lifted-PSD Stability… · p. 3

B. Certified Safe Set and Sample-Safe Radius

The certified safe set is defined by target-branch physical feasibility together with a fixed lifted stability certificate, and sample-safe radii are validated through nested set inclusions.

Safe-set definition

  • The physical safe set evaluates operating limits, target-model dynamics, and PF-Jacobian regularity on the target AC-PF branch.The target branch is denoted xPF(u, ξ), with JxF and the spectral abscissa evaluated at its equilibrium.Source: B. Certified Safe Set and Sample-Safe… · p. 3

Certification scope

  • The certified safe set is an inner approximation of the physical safe set, Scert(u) ⊆Sphys(u), under the specified target model.A higher-fidelity interpretation requires a separate uniform bridge covering equilibria, regularity, limits, and dynamics.Source: B. Certified Safe Set and Sample-Safe… · p. 3

Sample-safe radius

  • A sample radius is certified by requiring its weighted ℓ1 ball to lie inside Scert(u), which implies 0 ≤ri ≤dcert i (u) ≤dphys i (u).The ball is propagated through the nonlinear PF map and enclosed by a tractable state tube and certified convex lifted set.Source: B. Certified Safe Set and Sample-Safe… · p. 3, Figure 1 · p. 4

C. Exact Wasserstein Risk Aggregation

The framework inserts certified sample-distance lower bounds into the exact distance-based Wasserstein reformulation, preserving tractable distributional risk aggregation while retaining certificate conservatism.

Wasserstein model

  • The empirical distribution is formed from realized-minus-forecast error samples, with a 1-Wasserstein ambiguity set using the fitted metric ∥δ∥D = ∥Dδ∥1.The data and transport remain in forecast-error space, while D sets directional transportation cost.Source: C. Exact Wasserstein Risk Aggregation · p. 4

Safety implication

  • Because Scert(u) ⊆Sphys(u), controlling certified failure also controls failure for the specified target model.The certified failure set is closed under the strict safe-set conditions.Source: C. Exact Wasserstein Risk Aggregation · p. 4

Risk aggregation

  • The exact reformulation aggregates the true sample distances through auxiliary variables t and si under the Wasserstein risk budget.Conditional on the true distances, the reformulation is exact.Source: C. Exact Wasserstein Risk Aggregation · p. 4

Certified-distance substitution

  • Certified lower bounds safely replace exact distances through ri ≥t −si, si ≥0, with the aggregate constraint ρ + Σi=1 si ≤αt.The replacement preserves a conservative Wasserstein guarantee because ri ≤dcert i (u).Source: C. Exact Wasserstein Risk Aggregation · p. 4

WDRO-OPF

  • The resulting WDRO-OPF minimizes operating cost subject to dispatch feasibility, t, si, and certified-radius constraints.Its conservatism comes from certified lower bounds and any gap between the fixed certificate and target-model stability.Source: C. Exact Wasserstein Risk Aggregation · p. 4

III. CERTIFIED SAMPLE-SAFE RADII

The section constructs sample-centered forecast-error balls whose nonlinear AC power-flow images remain certified safe through branch-preserving PF tubes and conic conditions.

Sample-ball construction

  • The Wasserstein requirement becomes a search for a radius around each empirical forecast-error sample whose entire ball remains inside the certified safe set.Each realization must pass through the nonlinear, branch-dependent AC power-flow equations before stability and operating conditions can be checked.Source: §III · p. 4
  • A PF anchor and fixed verified coefficients define the sample-centered tube and the online variables updated by the conic master.The coefficients, Perron scalings, adjoint quantities, and remainder bounds are held fixed while dispatch, radius, and tube variables are updated.Source: §III · pp. 4–5

PF representation

  • The AC equations are represented exactly as affine-quadratic expressions in state, dispatch variation, and forecast-error variation.The retained center residual prevents a floating-point PF center from being treated as an exact root.Source: A. Branch-Preserving Component-Perron… · p. 5

Branch-Preserving Component-Perron PF Tube

  • Componentwise self-mapping and Perron-scaled contraction conditions certify a unique, continuous, Jacobian-regular PF solution throughout each weighted-ℓ1 sample ball.The target-branch label propagates from the certified center root across the ball under the stated initialization or continuation condition.Source: A. Branch-Preserving Component-Perron… · p. 5
  • Banach contraction and nonsingularity arguments establish existence, uniqueness, continuous parameter dependence, and a nonsingular PF Jacobian throughout the certified tube.The conclusion relies on the verified matrix inequality, self-mapping bounds, and contraction condition.Source: A. Branch-Preserving Component-Perron… · p. 5
  • The certified tube remains on the positive voltage-magnitude branch in the Iva specialization through an additional lower-bound condition.These applicability restrictions are imposed as extra tube or safety-block conditions.Source: A. Branch-Preserving Component-Perron… · p. 5

B. Exact Adjoint Safety-Matrix Identity

The adjoint construction eliminates the linear state dependence from safety outputs exactly on the PF graph, leaving affine terms plus quadratic state remainders for unified stability and operating-limit treatment.

Exact adjoint identity

  • Each lift coordinate used by stability and operating-limit blocks is represented through an exact block identity.The construction includes case-specific linear coordinates without approximation.Source: B. Exact Adjoint Safety-Matrix Identity · pp. 5–6
  • Adjoint systems are solved for the distinct safety-output directions to remove the first-order state term without approximating the nonlinear PF graph.Only distinct lift directions appearing in the safety blocks require adjoint right-hand sides.Source: B. Exact Adjoint Safety-Matrix Identity · p. 6
  • On the PF graph, the resulting safety outputs have an exact affine-quadratic representation.The proof substitutes the PF equality into the adjoint identity to obtain the stated expressions.Source: B. Exact Adjoint Safety-Matrix Identity · p. 6
  • The adjoint step removes only first-order state dependence and leaves an exact quadratic residual.It therefore does not replace or linearize the nonlinear PF equations.Source: B. Exact Adjoint Safety-Matrix Identity · p. 6

Unified safety blocks

  • Stability, scalar upper limits, and SOC limits are assembled as affine symmetric PSD blocks indexed by safety block κ.The implementation assembles these blocks blockwise rather than forming one dense matrix.Source: B. Exact Adjoint Safety-Matrix Identity · p. 6
  • Combining output directions before bounding the operator-valued quadratic remainder preserves cancellations lost under independent lift-coordinate enclosures.This provides a tighter matrix-level treatment of the shared nonlinear variation.Source: B. Exact Adjoint Safety-Matrix Identity · p. 6

C. Finite Matrix-Remainder Robust Containment

The matrix-remainder construction replaces nonlinear safety variation with finite robust PSD constraints using matrix-level certificates and exact weighted-ℓ1 vertex handling.

Finite robust containment

  • Fixed signed output majorants and offline matrix variables produce domination LMIs for the nonlinear safety remainder.The majorants remain offline parameters to avoid bilinear products with online variables.Source: C. Finite Matrix-Remainder Robust… · pp. 6–7
  • A retained state-interaction core graph and analytically bounded tail exploit shared quadratic state factors in the safety blocks.The resulting PSD block matrix is a degree-two Gram certificate within a matrix sum-of-squares framework.Source: C. Finite Matrix-Remainder Robust… · p. 6
  • The remainder is split into coordinate-box and matrix-shaped parts, each bounded by finite PSD-compatible terms.The coordinate-box contribution uses summed matrix variables, while the matrix-shaped contribution is bounded by a matrix function T_iκ.Source: C. Finite Matrix-Remainder Robust… · pp. 6–7
  • Weighted-ℓ1 affine uncertainty is handled exactly through its 2m signed vertices.The signed coordinate-box and pure matrix-remainder certificates are recovered by setting the split coefficients to (1, 0) or (0, 1).Source: C. Finite Matrix-Remainder Robust… · p. 7
  • With offline quantities fixed, the resulting conditions are linear, SOC, or LMI constraints in online variables, with no semi-infinite constraint remaining.This yields a finite conic robust counterpart for the certified containment problem.Source: C. Finite Matrix-Remainder Robust… · p. 7

D. Certified-Radius Guarantee

Theorem 2 composes PF regularity, exact safety identities, and matrix-remainder bounds to certify a sample-safe radius as a lower bound on sample-to-failure distance.

Certified-radius guarantee

  • Theorem 2 requires the model-specific stability certificate, PF branch conditions, verified adjoints, numerical bounds, and all safety-block inequalities to hold.These hypotheses are imposed uniformly over the certified domain.Source: D. Certified-Radius Guarantee · p. 7
  • For every forecast-error realization in the sample ball, Theorem 1 supplies a unique regular target-branch equilibrium within the certified state tube.The exact adjoint identities then evaluate every safety block at that equilibrium.Source: D. Certified-Radius Guarantee · p. 7
  • The finite robust LMIs imply every stability and operating safety block is PSD throughout the ball.The argument uses the convex-hull representation of the weighted-ℓ1 uncertainty set.Source: D. Certified-Radius Guarantee · p. 7
  • The ball is contained in the certified safe set, which lies inside the physical safe set under the model-specific certificate theorem.Consequently, the closed ball cannot intersect the closed failure set, yielding a certified lower bound on sample-to-failure distance.Source: D. Certified-Radius Guarantee · p. 7
  • Exact PF expansions, adjoint identities, and weighted-ℓ1 vertex reduction are nonconservative, while conservatism enters through PF tubes, matrix-remainder bounds, and sufficient stability certificates.Theorem 2 composes these elements into the radius required by the Wasserstein risk interface.Source: D. Certified-Radius Guarantee · p. 7

A. Convex Master With Fixed Local Bounds

With fixed local bounds, the master problem jointly handles certified radii, nominal and sample safety conditions, and dispatch within a trust region. PF branch-label propagation supplies the regular target branch, but the sequential scheme is not globally optimal.

Convex master

PF branch certification

  • Componentwise self-mapping and Perron-contraction conditions preserve a unique, continuous, locally smooth, regular PF path between labelled roots.The proof uses affine interpolated inequalities, Banach’s theorem, strict interiority, and the implicit-function theorem.Source: A. Convex Master With Fixed Local Bounds · pp. 7–8
  • Endpoint root enclosures can inherit the target-branch label when they lie within the previously certified endpoint tube.Source: A. Convex Master With Fixed Local Bounds · p. 8

Scope

B. Acceptance and Distributional Guarantee

Candidates are accepted only after independent verification of PF, safety, Wasserstein, and branch conditions; otherwise the trust region shrinks and local bounds are recomputed.

Acceptance

  • Independent PF solves verify all safety and Wasserstein conditions and Proposition 1 before accepting a candidate iterate.If verification or direct continuation is inconclusive, the trust region is reduced, bounds are recomputed, and verified continuation may be used.Source: B. Acceptance and Distributional… · p. 8

A. Mechanism Study: Certified Radius Versus First Failure

The two-bus study compares the certified radius with a closed-form first target-model failure distance and examines the associated stability transition. The certificate is tight while remaining strictly inside the failure boundary.

Mechanism study

  • The two-bus lossless GFM experiment perturbs active-power transfer with renewable and load forecast errors measured in the Wasserstein weighted-ℓ1 metric.Both voltage magnitudes are fixed at 1 p.u.Source: A. Mechanism Study: Certified Radius… · p. 8
  • The critical transfer is Pcrit = 0.602 p.u., and the nearest weighted-ℓ1 failure distance is available in closed form.The projected Iva matrix is positive definite precisely on the certified stable side.Source: A. Mechanism Study: Certified Radius… · p. 8
  • The first event is loss of small-signal stability because the PF Jacobian remains regular and operating limits retain positive margins at dfirst.The opposite transfer direction reaches its boundary at a larger distance.Source: A. Mechanism Study: Certified Radius… · p. 8
  • 0.03993374 is the certified radius, capturing 99.8344% of the exact distance with 0.1656% relative conservatism.The radius is therefore immediately inside the first-failure boundary rather than coinciding with it.Source: A. Mechanism Study: Certified Radius… · p. 8
  • At normalized distance one, the Iva minimum eigenvalue and target-model spectral abscissa cross zero together while the PF branch remains regular.At 0.9999 they are 2.41 × 10^-8 and −1.96 × 10^-7; at 1.0001 they are −2.41 × 10^-8 and 1.96 × 10^-7.Source: A. Mechanism Study: Certified Radius… · p. 8
  • Nonlinear modal-growth fits agree with linearized spectral abscissae within 4.21 × 10^-6, with coefficients of determination above 0.99999989.Perturbations decay on the stable side and grow on the unstable side for both signs of the dominant mode.Source: A. Mechanism Study: Certified Radius… · p. 8
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