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Minimum Bounding Sets for N-1-1 Branch Contingencies Under Thermal and Voltage Limits in Linear Network Models

Timothy N. Thacker

arXiv:2610.12606v1eess.SYcs.SY

TL;DR

Traditional single-outage rankings cannot guarantee that omitted N-1-1 pairs are bounded. This paper derives exact minimum bounding sets for thermal and voltage criteria in linear network models using outage-coupling bounds and a best-first search. On public systems, the search reproduces brute-force worst cases with far fewer evaluations, while its guarantees do not extend to the nonlinear AC equations.

  • Problem

    Traditional ranking-based screening cannot guarantee bounds on omitted outage pairs, whose number grows quadratically with eligible elements.

    Source: §I · p. 1

  • Method

    The paper bounds pair interactions using outage coupling and applies a best-first search to linear thermal and voltage models.

    Source: §I · p. 1, §X · p. 10

  • Results

    On public systems of up to 2,869 buses, the search reproduced brute-force worst cases with up to 3,800 times fewer pair evaluations for thermal criteria and 23,700 times fewer pair–bus evaluations for voltage criteria.

    Source: §X · p. 10

  • Takeaways & Limitations

    The resulting sets are exact for the linear models, but for AC they are strong candidate lists rather than guaranteed bounds.

    Source: §X · p. 10

    The bounds are properties of linear models; for nonlinear AC, the sets are strong candidate lists but cannot be guaranteed.

    Source: §IX · pp. 9–10

Abstract

from arXiv · show

N-1-1 contingency analysis studies the loss of one element followed by system adjustments and the loss of a second. The number of contingency pairs increases quadratically with the number of applicable elements, and traditional ranking-based screening methods cannot guarantee that omitted pairs are bounded. This paper determines the minimum bounding set of two-branch-outage pairs: the smallest set that can reproduce the worst value of every violated thermal and voltage limit of a linear network model. In the DC model, effects of an outage pair can be quantified by the extent of the coupling between the outages. The paper measures this by the cosine between the two outages in the network's cycle space, with a bound that splits into one term per outage and is within a near-ideal factor of $\sqrt{1+\varepsilon}$. This argument extends to every linear model in which an outage subtracts a positive semi-definite low-rank term from a symmetric positive definite matrix, with a tight bound and a collapse test. A model with voltage magnitudes is extracted by performing one Newton step on the energy function of the lossless network in rectangular coordinates. A best-first search built on these bounds reproduced brute force on public systems of up to 2{,}869 buses with 130 to 3{,}800 times fewer pair evaluations for flows and 170 to 23{,}700 times fewer for voltages. Gaps in translation to nonlinear models are explored on the IEEE 118-bus system.

I. INTRODUCTION

N-1-1 screening must identify severe outage pairs without exhaustively studying a quadratic number of sequences, while traditional single-outage rankings cannot bound what they omit. The paper defines exact minimum bounding sets for linear-model thermal and voltage criteria and outlines the methods and scope used to obtain them.

I. INTRODUCTION

  • Traditional rankings based on single-outage severity cannot bound omitted pairs, despite the |E|(|E| −1) ordered sequences created by |E| eligible elements.The paper frames its question as finding the smallest defensible set whose study bounds every other pair without full enumeration.Source: §I · p. 1
  • The exactness claim is limited to two-branch outages in linear models: DC for thermal criteria and a symmetric linear AC model for voltage criteria.System adjustments between outages are included for thermal evaluation.Source: §I · p. 1

A. Related Work

  • Prior N-2 screening uses LODF-derived bounds to filter pairs and reports no missed violating pairs in the DC model [3]–.Related work also describes interference between outage responses and limitations of single-outage ranking.Source: A. Related Work · p. 1

B. Contributions

  • The contributions include a cycle-space interaction bound, an exact best-first search, extensions to low-rank outage models and voltage magnitudes, and studies of adjustment and AC-model gaps.The minimum bounding set contains a maximizer of each violated criterion, accounting for ties through a minimum hitting set.Source: B. Contributions · pp. 1–2, §II · p. 2

II. PROBLEM FORMULATION

  • A set bounds a criterion family when it reproduces the maximum of every violated criterion across the finite study cases.The set must be recalculated after corrective actions change the network.Source: §II · p. 2

III. OUTAGE ALGEBRA IN THE CYCLE SPACE

The DC pair-outage response is expressed in cycle-space quantities, where the cosine between outages captures their coupling. A separable bound controls the interaction beyond the two single-outage changes and is nearly tight.

A. Known Pair Algebra

  • The cycle-space formulation expresses pair effects using the Gram matrix of the outage vectors, whose off-diagonal cosine measures coupling.The absolute cosine also equals the square root of the collective-effects predictor κ_kj.Source: A. Known Pair Algebra · p. 2

B. The Interaction Bound

  • The section’s new bound controls the part of the pair change not quantified by the two single-outage changes.Its split form supplies separate terms for the two outages, as required by the search.Source: B. The Interaction Bound · p. 2
  • For coupling cosine bounded by ε, the tight constants satisfy C*_2 = ε/√(1 − ε) and ε/√(1 − ε²) ≤ C*_1 ≤ ε/√(1 − ε).The first constant is attained in the stated vector-configuration setting; extremal configurations are not shown to arise from positive-susceptance networks.Source: B. The Interaction Bound · pp. 2–3
  • The four-bus example raises monitored flow from 1.7 to 30.0 MW after k then j, although neither single outage changes that flow.This illustrates hidden amplification from coupled outages.Source: Figure 1 · p. 3

C. How Many Pairs Are Strongly Coupled

The coupling budget limits how many strongly coupled partners an element can have when those partners lie on cycles, but partners with little cycle exposure escape that count.

C. How Many Pairs Are Strongly Coupled

  • Partners lying on almost no cycle escape the coupling budget, and they are numerous on the small test systems.The result therefore does not provide a count bound for all strongly coupled pairs without qualification.Source: C. How Many Pairs Are Strongly Coupled · p. 3

D. Elements That Trip Several Branches

The interaction bound extends from individual branches to elements that trip several branches through a subspace coupling measure. Single-outage rankings still cannot bound all pair effects because outage interactions can amplify, attenuate, or reverse individual impacts.

D. Elements That Trip Several Branches

  • For multi-branch elements, the coupling cosine becomes the cosine of the smallest principal angle between their cycle-space subspaces.On about 3,900 random two-branch-element pairs in each of three smaller systems, the largest interaction-to-bound ratio was 0.998.Source: D. Elements That Trip Several Branches · p. 3

E. Why Ranking Rules Cannot Bound

  • Single-outage rankings cannot bound pair effects because two outages can amplify, attenuate, or reverse their individual impacts.In the example, losing k and j forces flow onto m even though neither single outage changes m’s flow.Source: E. Why Ranking Rules Cannot Bound · p. 3

IV. BEST-FIRST SEARCH

The best-first search combines outage-pair bounds to find the exact worst violating pair without evaluating every pair. Its stopping rule is instance-optimal with respect to the information in those bounds.

A. Bounds and Algorithm

  • The bound for a strong pair is within a factor of √(1+ε) of optimal.For strong pairs, the cap is the smaller of two outage-based bounds.Source: A. Bounds and Algorithm · p. 3
  • A stress-based bound rules out pairs involving near-radial branches, whose coupling ξ_m is small.The bound uses the two largest outage stresses.Source: A. Bounds and Algorithm · p. 3
  • The weak-pair stream enumerates pairs by decreasing summed scores, and the merged streams release pairs in decreasing bound order.Pairs leave the search in decreasing order of their bound.Source: A. Bounds and Algorithm · p. 3
  • Algorithm 1 returns the exact maximum and a maximizing pair when a violation exists, and reports none otherwise.It evaluates every pair whose bound exceeds the larger of the maximum and the criterion threshold, plus possibly ties.Source: A. Bounds and Algorithm · pp. 3–4
  • The returned worst values do not depend on ε, which changes only the number of evaluated pairs and is not a screening threshold.Continuing the search with the stated stopping conditions can collect tied maximizers or list every violating pair.Source: A. Bounds and Algorithm · p. 4

B. A Priori Work Bound

The section gives an a priori count for weak-pair evaluations and extends pair-outage bounds to linear models with low-rank outage terms. The general bound is tight, and the model also admits a test for whether a pair makes the system singular.

B. A Priori Work Bound

  • Under the stated work-bound conditions, Algorithm 1 evaluates at most N(N−1)/2 weak pairs, with N=⌊A²/(µ−ϱ)²⌋.A tighter pre-search count uses the elements whose individual scores exceed the specified criterion-dependent cutoff.Source: B. A Priori Work Bound · p. 4

V. VOLTAGE CRITERIA: LINEAR MODELS WITH LOW-RANK OUTAGES

  • The voltage criteria use a symmetric positive-definite linear model in which each outage removes a positive-semidefinite low-rank branch term.The pair value is evaluated against the base voltage threshold, assuming criteria already violated in the base case are excluded.Source: §V · p. 4

A. Low-Rank Outage Models

  • A pair requires one solve of order r_k+r_j, and an inertia count on the same matrix tests whether the pair collapses the model.The pair-state change is computed from the low-rank update when the corresponding matrix is nonsingular.Source: A. Low-Rank Outage Models · p. 4
  • For a general low-rank outage model, pair interaction is bounded using the coupling c/(1−c), and no smaller constant works for all outage stresses and signals.The pair formula uses the exposures, outage stresses, and coupling between the two branch terms.Source: A. Low-Rank Outage Models · p. 4
  • The DC model has a sharper interaction bound because its normalized branch factors are orthonormal in cycle space; the general model lacks that Gram structure.For general models, the paper states that the tight interaction constant cannot be improved.Source: A. Low-Rank Outage Models · p. 5

B. A Symmetric Linear AC Model

The paper constructs a voltage-magnitude model by taking one Newton step on the lossless-network energy function in rectangular coordinates. This yields a low-rank outage model, while the paper identifies limitations in alternative linear voltage models.

B. A Symmetric Linear AC Model

  • One Newton step on the energy function produces a low-rank outage model for voltage magnitudes and linearized active and reactive branch-flow criteria.Voltage change at a PQ bus is represented as a linear functional of the state displacement.Source: B. A Symmetric Linear AC Model · p. 5
  • The model matrix is positive definite exactly when the lossless power-flow Jacobian is, and it remains positive definite along a loading path up to the first saddle-node bifurcation.The equivalence follows from a nonsingular coordinate transformation of the symmetric Jacobian.Source: B. A Symmetric Linear AC Model · p. 5
  • Rectangular coordinates make the network energy exactly quadratic, placing the released outage injection in the range of the branch term.The two traditional linear voltage models each lack one of the properties needed for the approach: symmetry, positive-definiteness, or range inclusion.Source: B. A Symmetric Linear AC Model · p. 5
  • Pairs with coupling c_kj≥1 are treated as strong and evaluated; for c_kj<1, the interaction bound applies.The search uses ω=ε/(1−ε) for weak-pair scores, and solves a pair only when it leaves the priority queue.Source: B. A Symmetric Linear AC Model · p. 5

VI. ADJUSTMENT BETWEEN CONTINGENCIES

With adjustment between outages, each first contingency is checked against a feasible adjustment polyhedron, and second-contingency constraints are added by constraint generation. A witness bound limits how many second contingencies determine failure or adjustment cost, but does not reduce the work needed to establish passing.

VI. ADJUSTMENT BETWEEN CONTINGENCIES

  • After the first outage, generation redispatch and other adjustments are constrained by a polyhedron encoding operational limits and power balance.Post-outage flows are affine functions of the adjustment.Source: §VI · p. 5
  • Constraint generation finds the most violated second-contingency row with Algorithm 1, while unreached rows are certified safe.The problems decouple over first contingencies.Source: §VI · p. 5
  • For an adjustment space of affine dimension d, at most d+1 rows certify infeasibility, while at most d rows preserve the feasible problem's optimal value.The bounds follow from Helly’s theorem and a Carathéodory reduction of the optimality conditions.Source: §VI · pp. 5–6
  • The witness bound does not reduce the work of showing a first contingency passes, and passing each pair separately need not imply a common feasible adjustment.Two second contingencies requiring u≥1/2 and u≤−1/2 each pass alone but fail together over U_k=[−1,1].Source: §VI · p. 6

VII. THE AC EQUATIONS: PER-PAIR CERTIFICATES AND OPEN GAPS

For AC equations, the paper gives a certificate for each evaluated outage pair, but no bound for excluding unevaluated pairs. It identifies limits in extending linear-model guarantees to nonlinear AC behavior.

VII. THE AC EQUATIONS: PER-PAIR CERTIFICATES AND OPEN GAPS

  • For an evaluated pair, the Newton–Kantorovich theorem gives a certificate when h = ϑη ≤ 1/2, bounding bus voltages and branch apparent power.The certificate uses a Newton step and a global Lipschitz condition for the quadratic mismatch map.Source: §VII · p. 6
  • No AC exclusion bound is provided, and the energy model has the required structure only for one Newton step from the base state.The remainder of the AC equations has no corresponding bound; the classical max-norm Kantorovich bound is vacuous on IEEE 118-bus.Source: §VII · p. 6
  • The paper does not bound differences between linear-model and AC values for unevaluated pairs, and pairs far from the base state can be understated.A guarantee for large deviations requires bounding the solution set over a region, rather than at one point.Source: §VII · p. 6
  • The per-pair AC guarantee requires one Jacobian factorization per pair and does not reduce the number of AC solutions.Reactive limits, tap changes, and switched shunts change the mismatch map, requiring a separate guarantee for each switching state.Source: §VII · p. 6

VIII. CASE STUDIES

Case studies test the bounds and best-first search on five public systems, with system loading and model settings varied. The search reproduces brute-force worst values while evaluating far fewer pairs.

VIII. CASE STUDIES

  • The five public test systems ranged from IEEE 118-bus to PEGASE 2869-bus, with non-bridge branches eligible and all branches monitored.Thermal studies used synthetic ratings for IEEE and PEGASE cases; ACTIVSg cases also used native ratings, and the evaluation comparison used ε = 0.02 by default.Source: §VIII · p. 6

A. Structure of the Pair Space

  • Strongly coupled pairs were a small fraction of the pair space; on PEGASE 1354, 6,904 of about one million pairs were strong at ε = 0.1.The budget identity held within 3 × 10^-12 on every system, while the theoretical count was loose by a factor of 50 to 100.Source: A. Structure of the Pair Space · p. 6
  • All sampled interaction terms lay below Proposition 1’s bound, whose largest sampled ratio reached 0.998 to 0.999 on the three smaller systems.For PEGASE 2869, the largest ratio was 0.65; fewer than 0.3% of sampled pairs had |c| > 0.1.Source: A. Structure of the Pair Space · p. 6, Figure 2 · p. 7

B. Exactness and Size of the Minimum Set

  • 893 sequences bound 1,616 violated criteria among approximately 7.2 million pairs on PEGASE 2869 at λ = 0.8.The search returned brute-force maxima for violated criteria and found no violation elsewhere; the minimum set is smaller than the 1,736 violating pairs because pairs can be worst for multiple branches.Source: B. Exactness and Size of the Minimum Set · p. 7, Table I · p. 7

C. Search Effort

  • 130 to 3,800 times fewer pair evaluations than brute force were needed, with the reduction growing by system size and shrinking as systems were stressed.The search matched vectorized brute force in wall-clock time on two small systems and was faster on the larger systems.Source: C. Search Effort · p. 7
  • Every tested ε returned identical worst-case values, while effort was lowest near ε = 0.01–0.02.Evaluations per searched criterion ranged from about 40 on the 118-bus system to about 2,000 on PEGASE 2869.Source: C. Search Effort · p. 7, Figure 4 · p. 7

D. Comparison With the Filter of [5]

The case studies compare the search with a filter, examine native ratings, dispatch and adjustment settings, and evaluate voltage criteria in the energy model. The search remains exact for the reported linear-model criteria, while its evaluation savings and the filter’s relative performance vary by setting.

D. Comparison With the Filter of [5]

  • At light loading the filter used fewer evaluations, but under stress the search used 24 to 83 times fewer on the listed systems and loadings.Counting bound computations, the search required 2.5 to 84 times fewer operations than the filter.Source: D. Comparison With the Filter of [5] · p. 8

E. Native Ratings, Several Study Cases, and Ranking Rules

  • On ACTIVSg2000 with native ratings at λ = 1.0, 673 sequences bounded 961 violated criteria, with about 2,200 times fewer evaluations than brute force.At λ = 1.1, 885 sequences bounded all criteria, and the search remained exact.Source: E. Native Ratings, Several Study… · p. 8
  • A rule pairing each branch’s ten most severe single outages missed worst sequences for 6 of 78, 40 of 120, and 51 of 657 violated criteria across three systems.For PEGASE 1354, one worst pair included an element ranked 1,428th of 1,429 by single impact.Source: E. Native Ratings, Several Study… · p. 8

F. Adjustment Between Contingencies

  • With redispatch on IEEE 118-bus, 23, 38, and 68 first contingencies failed at λ = 0.8, 0.9, and 1.0; load interruption secured all but 4, 7, and 17 failures.Redispatch controls were 54 generators limited to 10% of capacity; interruption was at most 144 MW for the reported secured cases.Source: F. Adjustment Between Contingencies · p. 8

G. Voltage Criteria

  • The voltage interaction bound held across roughly 2.4 × 10^10 pair–bus combinations, with the largest interaction-to-bound ratio from 0.96 to 0.9999996.Most pair coupling was weak; c_kj ≥ 1 occurred only for pairs flagged by the collapse test.Source: G. Voltage Criteria · p. 8
  • 170 to 23,700 times fewer pair–bus evaluations were needed for voltage criteria, and the search reproduced brute-force worst values and tie sets.The reported minimum sets contained 25 to 532 sequences for 34 to 1,205 violated criteria; these sets bound the energy model, not AC.Source: G. Voltage Criteria · p. 8, Table II · p. 9

H. The AC Equations

The AC study finds that the linear-model guarantees can produce exact candidate sets for the tested criteria, but those guarantees do not extend to AC values. The conclusion reports large reductions in evaluations for linear models while identifying an AC remainder bound as a route toward certifying omitted pairs.

H. The AC Equations

  • The per-pair guarantee held for all but one tested pair, and its guaranteed radius exceeded actual error by a median of 1–2% and never more than a factor of two.The guarantees alone reproduced AC minimum sets of 41 and 77 thermal sequences and 12 and 22 low-voltage sequences; the unguaranteed pair indicated voltage collapse as Newton’s method failed.Source: H. The AC Equations · p. 9
  • The DC worst pair ranked first in the AC ordering for 60 of 67 and 109 of 136 violated branches, and within the top five for all but one.Where the DC pair differed, it could understate the AC worst loading by up to 21%; the exception at DC rank 168 was driven by voltage depression rather than active-power redistribution.Source: H. The AC Equations · p. 9
  • The energy voltage model missed 6 of 53 AC-violated criteria, versus 10 for power-mismatch linearization and 26 for the decoupled model, while flagging 13 criteria AC did not violate.Its correlation with AC voltage changes was 0.97 to 0.98; half its misses involved the bus 5–8 transformer outage, whose strongly nonlinear response also caused power-mismatch linearization to fail.Source: H. The AC Equations · p. 9

IX. DISCUSSION AND LIMITATIONS

  • The linear bounding sets are not AC bounds, and no current bound relates unevaluated pairs’ linear predictions to their AC values.The paper proposes bounding the AC remainder using energy-model pair quantities, with the guarantee certifying pairs not excluded.Source: §IX · pp. 9–10
  • The study covers branch outages; its AC analysis omits reactive limits, tap changes, and switched shunts, while adjustment effects were tested on only one system.The energy model drops conductances, and a symmetric bound for a lossy version remains open.Source: §IX · p. 10

X. CONCLUSION

  • Up to 3,800 times fewer pair evaluations for thermal criteria and up to 23,700 times fewer pair–bus evaluations for voltage criteria reproduced brute-force worst cases on public systems up to 2,869 buses.Against nonlinear AC models, these sets are strong candidate lists rather than guaranteed bounds; a remainder bound is proposed as a route to excluding AC pairs without evaluation.Source: §IX · pp. 9–10, §X · p. 10

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