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Recovering Plausible Transmission Outage Scenarios

Karen Ji, Cameron Khanpour, Samuel Talkington, J. Kyle Skolfield, Daniel K. Molzahn

arXiv:2610.29872v1eess.SYcs.SYmath.OC

TL;DR

Public outage records often report customers without power by county, not which transmission branches failed, complicating construction of hazard-related outage scenarios. The paper matches regional load-shed observations with branch-outage scenarios using a bilevel DC OPF and sensitivity-based search. The recovered scenarios leave 0.02%–4.52% of E0 unexplained on three synthetic systems and 23.12% on CATS for a December 2025 storm.

  • Problem

    Public records provide regional outage data without identifying the failed branches, leaving unclear how to construct hazard-event outage scenarios from those observations.

    Source: §I · p. 1

  • Method

    A bilevel formulation matches observed regional load shed with DC-OPF simulations, using lower-level sensitivities to search branch states and randomized rounding to form scenarios.

    Source: §I · pp. 1–2

  • Results

    0.02%–4.52% of E0 remains unexplained on three synthetic systems, and 23.12% remains unexplained on CATS for a December 2025 storm.

    Source: §VI · p. 10

  • Takeaways & Limitations

    Because several distinct scenarios can fit regional data similarly, planning studies should use a ranked collection rather than treat one scenario as the true outage.

    Source: §VI · p. 10

    Regional load shed constrains outage scenarios only loosely: best-fitting scenarios can share few lines with the true outages.

    Source: §VI · p. 10

Abstract

from arXiv · show

Expansion planning for extreme weather needs outage scenarios that specify which transmission branches fail. However, public outage records often only report customers without power by county, making it unclear how to exploit this information to create outage scenarios. We propose a method to recover sets of outaged branches that most closely match regional load shed data. This gives us sets of plausible outages for a given hazard event. We formulate a bilevel problem with a load shedding DC optimal power flow in the lower level. We relax the binary branch states to differentiate the lower level problem, replace it with its sensitivity matrix, and use gradient descent to update the upper level. We treat the relaxed branch states as probabilities and perform randomized rounding to restore integrality. On three test systems with synthetic outages, the recovered sets reduce the load shed error to $0.02-4.5\%$. On the California Test System with storm data from December 2025, the error falls to $23\%$. These scenarios give expansion planning a reliable source of outage data tied to observed hazards.

I. INTRODUCTION

Public outage data report regional customer outages but generally do not identify failed transmission branches, leaving planners without a clear way to build hazard-specific scenarios. The paper recovers and ranks branch-outage scenarios by matching observed regional load shed, using a bilevel DC OPF formulation and a sensitivity-based search.

I. INTRODUCTION

  • County outage records and storm records provide regional load shed and hazard footprints, but public sources do not identify which transmission branches failed.Detailed component-level outage records are generally not publicly available.Source: §I · p. 1
  • The method returns a ranked collection because distinct branch-outage sets can produce nearly the same regional load shed.The authors suggest planning studies draw from the collection rather than treating one scenario as the true outage.Source: §I · pp. 1–2
  • The recovery is a bilevel problem: binary upper-level branch states match observed regional load shed, while a lower-level DC OPF minimizes load shed.The objective matches simulated and observed regional load shed.Source: §I · p. 2
  • The paper motivates an all-candidate sensitivity search because a mixed-integer reformulation adds complementarity binaries and a greedy search requires repeated DC OPF solves.The California Test System has 4,313 candidate branches.Source: §I · p. 2
  • The search relaxes branch states, computes lower-level load-shed sensitivities by implicit differentiation, and updates the upper level with gradient descent before randomized rounding.Outage probabilities depend on how far the search moves each branch from initialization.Source: §I · p. 2
  • Across three synthetic-outage test systems and CATS using observed outages from a December 2025 storm, many distinct outage scenarios fit the same regional data, supporting scenario collections in planning studies.Source: §I · p. 2

II. PROBLEM FORMULATION

The formulation uses regional load-shed observations and hazard footprints to infer which candidate branches failed at one point during an event. A bilevel model selects branch states to match those observations while a DC OPF dispatches generation and sheds load.

II. PROBLEM FORMULATION

  • The model seeks branch-outage sets that explain observed regional load shed for one hazard event at one point in time.The lower level dispatches generation and sheds load subject to DC power flow, generator limits, and branch flow limits.Source: §II · p. 2

A. Network and Data

  • Each region has an observed load-shed target in per unit, capped at the demand assigned to that region so every target is attainable.The network is partitioned into regions, each containing a subset of buses.Source: A. Network and Data · p. 2
  • The risk set includes lines with at least one endpoint in a storm region, while transformers are excluded.Restricting the search to the hazard footprint reduces the number of decision variables.Source: A. Network and Data · p. 2
  • The assumptions attribute all observed load shed to risk-set branch outages and keep generators, transformers, and out-of-footprint lines in service.The DC model omits reactive power, voltage limits, and losses; the observation and modeled demand represent one point in time.Source: A. Network and Data · p. 2
  • The overview depicts public regional load-shed and storm-footprint data feeding the method, which returns ranked outage scenarios.Its example shows the three best CATS scenarios for a December 24, 2025 storm, with outaged lines bolded and storm counties shaded.Source: Figure 1 · p. 3

B. Bilevel Formulation

  • Each branch state is 1 when in service and 0 when outaged; the lower-level DC OPF scales branch susceptance and flow limits by that state.The OPF includes generator outputs, load shed, voltage angles, and branch flows; it fixes a reference angle.Source: B. Bilevel Formulation · p. 3
  • The upper level minimizes squared error between simulated and observed regional load shed, including regions with no observed outages.The formulation has between 71 and 4,313 binary upper-level variables.Source: B. Bilevel Formulation · p. 3
  • The paper reports fit as E(z)/E0, the share of squared observed load shed left unexplained; many scenarios can reach nearly the same error.The method therefore returns a ranked collection of low-error scenarios for each event.Source: B. Bilevel Formulation · p. 3

III. SOLUTION METHOD

The method searches a continuous relaxation rather than enumerating every binary branch topology. Multiple randomized trials use projected gradient descent, then randomized rounding to produce and rank outage scenarios.

III. SOLUTION METHOD

  • Exhaustive topology search is impractical because the problem has 2^|Erisk| candidate topologies and each evaluation requires a DC OPF solve.The authors note this is impractical even on the RTS-GMLC network.Source: §III · p. 3
  • Each Monte Carlo trial starts from a randomized warm start, runs projected gradient descent on relaxed branch states, and rounds its best relaxed solution into outage scenarios.The relaxation, sensitivities, warm start, and rounding rule are specific to this recovery problem.Source: §III · p. 4
  • The method merges and ranks scenarios from all trials; step-size rules and restarts safeguard the search against a nonsmooth objective.The paper lists the safeguard parameters in Table I.Source: §III · p. 4

A. Continuous Relaxation

The method relaxes branch states to make load shed responsive to optimization and uses load-shed sensitivities for projected gradient descent. Its sensitivity model is locally exact only while the active binding constraints remain unchanged.

A. Continuous Relaxation

  • Branch states z_j are relaxed to [z_off, 1], with z_off = 10^-4, and scale both branch susceptance and flow limit.The lower bound keeps the weighted reduced Laplacian nonsingular; intermediate states provide an optimization path but have no physical meaning.Source: A. Continuous Relaxation · p. 4
  • 91.6% of CATS risk-set branches kept their full rating at z_j = 0.1 when only susceptance was scaled, limiting load-shed response and search movement.At trial warm starts, approximately 97% kept their full rating.Source: A. Continuous Relaxation · p. 4

B. Sensitivities and Projected Gradient Descent

  • The load-shed sensitivity matrix gives the change in bus-level load shed per unit change in branch state, with susceptance and capacity contributions computed by implicit differentiation.The capacity contribution is nonzero only where a flow limit binds; differentiation assumes a nonsingular linear system, with a 10^-10 perturbation added when needed.Source: B. Sensitivities and Projected… · p. 4
  • The gradient is RMS-normalized over the risk set so the pre-clipping RMS branch move is α, with individual moves clipped at Δ_max = 0.25.This normalization applies to risk sets from 71 branches in RTS-GMLC to 4,313 in CATS.Source: B. Sensitivities and Projected… · p. 4
  • The sensitivity matrix is exact only while the binding constraints stay fixed, so long steps can cross flow-limit kinks and leave the region where the gradient describes the objective.The step-size procedure accounts for this limitation.Source: B. Sensitivities and Projected… · p. 4

C. Warm Start

The warm start places branches near their individual flow-limit binding points, while margin selection calibrates initial load shed. Backtracking, adaptive step selection, and restarts manage progress when the objective is nonsmooth or search stalls.

C. Warm Start

  • The warm start places each branch near the state where its own flow limit binds, based on its intact-network flow and loading.A margin and independent jitter are applied, while a floor keeps lightly loaded branches away from z_off.Source: C. Warm Start · p. 4
  • The margin is chosen by a short network-specific sweep so initial load shed is within an order of magnitude of the observed total.For RTS-GMLC and IEEE 300, initial load shed remains below the observed total even at the stated margins, so the method uses 0 and 0.01, respectively.Source: C. Warm Start · p. 4

D. Step Size and Restarts

  • Backtracking line search accepts steps using the Armijo condition and halves α for up to three trial steps, each requiring one DC OPF solve.If none passes, the method accepts a decreasing-error step when the decrease exceeds τ; otherwise it stays at the current iterate.Source: D. Step Size and Restarts · pp. 4–5
  • The next search's starting step grows when the prior step's linear-model prediction agrees sufficiently, but has a floor of α_0/4.This floor prevents the starting step from collapsing near a kink; a resumed search can still take shorter steps.Source: D. Step Size and Restarts · p. 5
  • Restarts address exhausted searches, stagnation, zero load shed and gradient, or infeasibility; each trial returns its best state and ends at T iterations or error below 0.05.Restart actions vary by condition, including perturbing branches, rebuilding the warm start, or moving toward the intact state to regain feasibility.Source: D. Step Size and Restarts · p. 5

E. Randomized Rounding

Randomized rounding converts relaxed branch states into outage scenarios using trip probabilities based on how far the search moved each branch from its start. Scenarios are sampled, evaluated, and returned as a ranked list.

E. Randomized Rounding

  • A branch moved from its start to z_off receives trip probability approximately 1, while a branch that did not move down receives probability 0.This bases outage probability on search movement rather than low relaxed values alone.Source: E. Randomized Rounding · p. 5
  • The m branches with the largest trip probabilities form the candidate pool, and 75 samples independently trip each pool branch according to its probability.The pool size m bounds the number of outages in any sampled scenario.Source: E. Randomized Rounding · p. 5
  • Distinct sampled scenarios are ranked by error, then outage count, then total-load-shed error; the five best per trial are merged and ranked across trials.The returned output is a ranked list of plausible outage scenarios.Source: E. Randomized Rounding · p. 5

1 Cap each ˆLc at the demand in region c

The supplied algorithm fragments show steps for initializing branch loading, updating the search, handling conditions, and collecting low-error outage scenarios.

1 Cap each ˆLc at the demand in region c

  • The algorithm solves the intact-network problem at z = 1 and computes branch loadings for branches in the risk set.Source: §1 · p. 6
  • During the search, it solves problem (1), applies Condition 4 and continues if infeasible, and computes g using equations (4)–(5).Source: §1 · p. 6
  • The algorithm draws 75 outage scenarios from the pool, solves problem (1) for each distinct scenario, and adds the five lowest-error scenarios to R.Source: §1 · p. 6

26 Merge identical scenarios in R and rank them

The synthetic-outage experiments test recovery of regional load shed from known outage scenarios across three systems. The best recovered scenarios leave at most 4.52% of the target load shed unexplained.

A. Setup

M Monte Carlo trials 5

  • Synthetic outages are generated by tripping electrically contiguous lines in one or two regions; the method receives only regional load shed and storm regions.The test systems are RTS-GMLC, IEEE 300, and WECC 240, with regions defined from their system areas or zones.Source: M Monte Carlo trials 5 · p. 6
  • Recovery is evaluated by relaxed error, the best scenario’s discrete error, trials reaching Etol, and overlap with the true outage.Overlap is diagnostic because the objective does not reward it.Source: M Monte Carlo trials 5 · p. 6

IV. RESULTS ON SYNTHETIC OUTAGES

  • Best recovered scenarios leave at most 4.52% of E0 unexplained on the synthetic outages.The experiments also assess overlap with the true outage, relaxed error, and convergence.Source: M Monte Carlo trials 5 · p. 7

B. Recovery Accuracy

On synthetic outages, the recovered scenarios closely match regional load shed, although they often share few lines with the true outages. Comparisons with rounding controls show that the relaxed solution generally yields lower discrete error.

B. Recovery Accuracy

  • Four of five RTS-GMLC trials reach Etol within eight iterations, while WECC 240 has four trials below 0.15 and one near 3.7.The relaxed search leaves at most 1.50% of E0 unexplained across the three networks.Source: B. Recovery Accuracy · p. 7
  • The best scenarios leave 0.02%, 1.38%, and 4.52% of E0 unexplained on RTS-GMLC, IEEE 300, and WECC 240, respectively.Their overlap with true outages is two of five, six of 27, and none of 20 lines.Source: Table IV · p. 7, B. Recovery Accuracy · p. 7

C. Value of the Relaxed Solution

  • Rounding the relaxed solution gives the lowest discrete error in every case except WECC 240 at m = 25.On IEEE 300, it reaches 7.1% versus 59.5% and 56.5%; on WECC 240 at m = 50, it reaches 4.5% versus 79.5% and 23.8%.Source: Table V · p. 7, C. Value of the Relaxed Solution · p. 7
  • On WECC 240 at m = 50, the warm-start control trips 47 of 50 pool branches, while the relaxed solution trips 11.The warm-start control assigns high trip probability to lightly loaded branches near the floor.Source: C. Value of the Relaxed Solution · p. 7

D. Effect of Restarts

Restarts improve stalled IEEE 300 trials, while the rounding pool size strongly affects discrete error. The best RTS-GMLC scenarios share some lines but differ otherwise, and the section introduces a California storm case study.

D. Effect of Restarts

  • Restarts lower four stalled IEEE 300 trials’ relaxed error from 1.42 to at most 0.304; at m = 12, the best discrete error improves to 0.179.At m = 50, the experiments reach similar discrete errors: 0.0347 with restarts and 0.0303 without.Source: D. Effect of Restarts · pp. 7–8, Table VI · p. 9

E. Effect of Pool Size

  • At m = 50, discrete error falls from 7.10% to 1.38% on IEEE 300 and from 91.48% to 4.52% on WECC 240; CATS falls from 62.28% to 23.12% at m = 200.Errors can flatten or rise beyond these pool sizes; the authors recommend sweeping m for each event.Source: E. Effect of Pool Size · p. 8, Table VII · p. 9

F. Several Plausible Outage Scenarios

V. CASE STUDY OF A CALIFORNIA STORM

  • The California storm case study applies the method to observed outage data on a geolocated network.Source: §V · p. 9

A. Data

The California case study maps county outage observations onto a geolocated synthetic grid and tests recovery against a December 2025 storm. The best CATS scenario leaves 23.12% of E0 unexplained, while the results support using a ranked collection rather than treating one scenario as the true outage.

A. Data

  • CATS has 59 regions and a risk set of 4,313 lines; the storm observations include outages in 33 counties, 12 outside the storm regions.The model converts customer counts to load shed using 41.6 kW per customer.Source: A. Data · p. 9

B. Results

  • Three of five CATS trials end between 15.5% and 16.2% of E0, and none reaches Etol within 25 iterations.Most of the decrease occurs in the first ten iterations.Source: B. Results · p. 9, Figure 4 · p. 10
  • With m = 200, the best CATS outage scenario trips 33 lines and leaves 23.12% of E0 unexplained.It sheds 5.28 p.u. of the 12.37 p.u. observed.Source: B. Results · p. 9
  • The five best scenarios at m = 100 contain 31 to 40 lines each and 86 distinct lines overall, concentrated from Humboldt County to Monterey County.They are densest around the San Francisco Bay Area and Monterey; 19 lines in the best scenario lie in the Monterey area.Source: B. Results · pp. 9–10, Figure 5 · p. 10

VI. CONCLUSION AND FUTURE WORK

  • On CATS with observed December 2025 storm outages, the best scenario leaves 23.12% of E0 unexplained; synthetic-system best scenarios leave 0.02% to 4.52%.Source: §VI · p. 10
  • Because distinct scenarios can reach similar error while sharing few lines with true outages, planning studies should use a ranked collection rather than treat one as the true outage.A sequence of observations from one event constrains the scenario more tightly than a single snapshot.Source: §VI · p. 10

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