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An Approach for Fast Cascading Failure Simulation in Dynamic Models of Power Systems

Sina Gharebaghi, Nilanjan Ray Chaudhuri, Ting He, Thomas La Porta

arXiv:2205.00103v1eess.SYcs.SYmath.DS

TL;DR

Detailed dynamic simulations are computationally expensive and not suitable for cascading-failure simulation. The paper proposes a predictor-corrector approach built on Backward Euler, reporting speedups over Trapezoidal-method simulation across three power-system benchmarks.

  • Problem

    Conventional dynamic simulations are computationally expensive and not suitable for running cascading-failure simulations.

    Source: B. Gaps in Literature · p. 2

  • Method

    The paper combines stiff-decay Backward Euler simulation with a predictor-corrector approach that uses eigendecomposition to address BEM hyperstability.

    Source: §VI · p. 15

  • Results

    ≈10×, ≈20×, and ≈35× speedups over Trapezoidal-method simulation were reported for the IEEE 118-bus system, IEEE 68-bus system, and Polish grid, respectively.

    Source: §VI · p. 15

  • Takeaways & Limitations

    The authors identify BEM-PC as particularly suited to longer-term cascading-failure simulations, during which BEM can use larger time steps than TM for most of the simulation.

    Source: C. Discussion on Computational… · p. 6

Abstract

from arXiv · show

The ground truth for cascading failure in power system can only be obtained through a detailed dynamic model involving nonlinear differential and algebraic equations whose solution process is computationally expensive. This has prohibited adoption of such models for cascading failure simulation. To solve this, we propose a fast cascading failure simulation approach based on implicit Backward Euler method (BEM) with stiff decay property. Unfortunately, BEM suffers from hyperstability issue in case of oscillatory instability and converges to the unstable equilibrium. We propose a predictor-corrector approach to fully address the hyperstability issue in BEM. The predictor identifies oscillatory instability based on eigendecomposition of the system matrix at the post-disturbance unstable equilibrium obtained as a byproduct of BEM. The corrector uses right eigenvectors to identify the group of machines participating in the unstable mode. This helps in applying appropriate protection schemes as in ground truth. We use Trapezoidal method (TM)-based simulation as the benchmark to validate the results of the proposed approach on the IEEE 118-bus network, 2,383-bus Polish grid, and IEEE 68-bus system. The proposed approach is able to track the cascade path and replicate the end results of TM-based simulation with very high accuracy while reducing the average simulation time by approximately 10-35 fold. The proposed approach was also compared with the partitioned method, which led to similar conclusions.

I. INTRODUCTION … C. Contribution of Our Work

The paper addresses the computational burden of dynamic cascading-failure simulation with a fast BEM-based approach designed to trace cascade paths and reproduce benchmark outcomes. It evaluates the approach against Trapezoidal-method simulations and identifies limitations in prior dynamic and hybrid approaches.

I. INTRODUCTION

  • Long-term dynamic cascade simulations require solving many nonlinear differential-algebraic equations, making statistical analysis difficult and motivating less accurate but more manageable QSS models.The paper’s objective is a fast approach that accurately traces cascade paths and supports statistical analysis.Source: §I · p. 1
  • The study assumes deterministic cascade evolution, excluding protective-relay mistripping and other malfunctions, unlike probabilistic approaches that allow multiple post-contingency trajectories,.The assumption is that system components act as expected throughout the cascade.Source: §I · p. 1

A. Literature on Dynamic Simulation of Cascading Failure

  • Dynamic cascading-failure literature spans review or proposal papers –, hybrid QSS–dynamic simulation, and dynamic-model approaches –, while the first group performed no cascade simulations.The review/proposal category includes simulator-development and parallelization proposals.Source: A. Literature on Dynamic Simulation of… · p. 1, B. Gaps in Literature · p. 2
  • Prior dynamic-model work found load modeling critical to cascade-risk evaluation and reported that DC QSS approximated early cascade paths but diverged from ground truth later.That study tested randomly selected N −2 contingencies.Source: A. Literature on Dynamic Simulation of… · pp. 1–2

B. Gaps in Literature

  • Dynamic cascade studies remain computationally burdensome; prior work simulated 88 of 1,200 candidate Polish-system cases as cascades, while hybrid switching may raise accuracy concerns and still begins with dynamic simulation.The cited study reported 1,081 cases without dependent outages and 31 that diverged.Source: B. Gaps in Literature · p. 2
  • Conventional planning simulations typically cover contingencies lasting up to 30 s and are computationally expensive, making their methods unsuitable for cascading-failure simulation.The paper attributes this mismatch to prior dynamic simulations using similar structures and integration methods as conventional planning models.Source: B. Gaps in Literature · p. 2

C. Contribution of Our Work

  • The method uses large time steps with implicit Backward Euler integration, a predictor-corrector approach to address hyperstability under oscillatory instability, and an adaptive COI frame that supports island formation.The adaptive center-of-inertia reference frame is intended to speed Newton iterations and work seamlessly when cascading failures form islands.Source: C. Contribution of Our Work · p. 2
  • The proposed approach achieves high accuracy and significant speedup on three power-system test networks with multi-tier cascades, and retains a significant speedup over a partitioned explicit-integration approach.The evaluated systems are the IEEE 118-bus system, IEEE 68-bus system, and 2,383-bus Polish network; the benchmark uses Trapezoidal-method simulation.Source: C. Contribution of Our Work · p. 2

II. DYNAMIC SIMULATION: PRELIMINARIES & STATE-OF-ART · A. Dynamic Simulation Preliminaries · B. Simultaneous Solution: State-of-art

The sections introduce power-system dynamic simulation as repeated solution of differential-algebraic equations (DAEs) across disturbance events, then review simultaneous implicit methods and their computational burden for cascading-failure studies.

II. DYNAMIC SIMULATION: PRELIMINARIES & STATE-OF-ART

  • The paper frames dynamic simulation preliminaries and simultaneous-solution methods around the challenge of using traditional power-system simulations for cascading failures.The dynamic model is represented by nonlinear DAEs, with discrete relay states capturing protection actions.Source: §II · p. 2, A. Dynamic Simulation Preliminaries · p. 2

A. Dynamic Simulation Preliminaries

  • Relay constraints govern breaker trips: violations start a countdown, and if they persist until its trip time, the relay changes the discrete breaker state.The change can alter the network admittance matrix and/or injected current; if the violation clears before the countdown ends, it stops.Source: A. Dynamic Simulation Preliminaries · p. 2
  • Cascading-failure simulation repeatedly solves an initial-value problem after events such as faults, line trips, and load shedding.The initial conditions include device states, bus voltages, and relay states.Source: A. Dynamic Simulation Preliminaries · p. 2
  • Partitioned methods solve algebraic and differential equations sequentially, while simultaneous methods combine them into nonlinear algebraic equations using implicit integration.Commercial software commonly uses explicit AB2 in partitioned methods; these methods are valued for flexibility and robustness but have numerical-instability drawbacks, whereas implicit simultaneous methods can use variable time steps and are widely explored in academic research.Source: A. Dynamic Simulation Preliminaries · pp. 2–3

B. Simultaneous Solution: State-of-art

  • Simultaneous state-of-the-art simulation typically uses Trapezoidal Method and Newton iterations to solve the discretized differential and algebraic equations [3, 13].The Newton process updates states and voltages using the Jacobian and stops when the infinity norm of the combined mismatch is at most the convergence tolerance.Source: B. Simultaneous Solution: State-of-art · p. 3
  • Despite sparse computation and variable time steps, TM-based state-of-the-art solvers remain computationally burdensome for cascading-failure simulation.The section also describes Newton variants, analytical or difference-based Jacobians, sparse direct or iterative linear solvers, and adaptive time steps based on local truncation error.Source: B. Simultaneous Solution: State-of-art · p. 3
  • At relay trips, the benchmark truncates the time step to the trip instant, updates the network and voltage solution, then reduces the step size before resuming integration.Its ground truth uses variable-step TM with Δt ∈ [0.002, 1] s and ϵ = 10−4, full Newton iterations, an analytical Jacobian, sparse calculations, and a 4th-order synchronous-generator model; simulation stops at convergence with no anticipated relay actions or at complete collapse.Source: B. Simultaneous Solution: State-of-art · pp. 3–4

III. PROPOSED METHODOLOGY FOR DYNAMIC SIMULATION OF CASCADING FAILURE · A. BEM: Absolute Stability and Stiff Decay Properties [13] · 1) Absolute stability property:

The methodology uses implicit Backward Euler (BEM) with an eigenanalysis-based predictor-corrector to support efficient cascading-failure simulation, alongside protection and islanding models. It prioritizes matching cascade paths and end states over exact trajectories, and analyzes BEM and Trapezoidal Method (TM) absolute stability.

III. PROPOSED METHODOLOGY FOR DYNAMIC SIMULATION OF CASCADING FAILURE

  • The simulation objectives are to match the ground-truth cascade path and end-state topology, voltage, frequency, and demand served while remaining computationally efficient.Exact state and algebraic-variable trajectories are acknowledged as elusive; the authors argue that accurate paths and end states can suffice for statistical analysis.Source: §III · p. 4
  • The proposed framework combines stiff-decay BEM with an eigenanalysis-based predictor-corrector to address BEM hyperstability, functional protection schemes, and an adaptive center-of-inertia frame for island formation.It models time-delayed overcurrent, local undervoltage load shedding, and first-swing out-of-step relays; the framework can accommodate other relay types.Source: §III · p. 4

A. BEM: Absolute Stability and Stiff Decay Properties [13]

  • BEM is derived by Taylor expansion centered at t_n+1 and is a first-order method.The passage states that discretizing the system equation with BEM yields an expression, but does not provide that expression here.Source: A. BEM: Absolute Stability and Stiff… · p. 4

1) Absolute stability property:

  • For the test equation ẋ = λx, absolute stability requires the numerical solution to satisfy |x_n+1| ≤ |x_n| within the stability region.The region is considered in the complex λΔt-plane.Source: 1) Absolute stability property · p. 4
  • TM's absolute stability region is the part of the λΔt-plane satisfying its amplification-factor condition AF ≤ 1, namely the left half-plane.Source: 1) Absolute stability property · p. 4
  • BEM's absolute stability region includes the entire left half-plane and the right half-plane outside the unit circle centered at (1, 0); both BEM and TM are numerically A-stable.Figure 1 depicts the two regions in gray.Source: 1) Absolute stability property · p. 4, Figure 1 · p. 4

2) Stiff decay property: … 1) Cascading Failure Simulation Subprocesses (a):

BEM’s stiff decay enables large steps that suppress fast oscillations, but hyperstability can hide oscillatory instability and divert cascade propagation. The proposed predictor-corrector approach addresses this issue within a variable-step BEM simulation workflow.

2) Stiff decay property:

  • BEM’s stiff decay lets large time steps suppress fast oscillations, whereas TM propagates fast-mode local errors across the simulation interval.As ℜ(λ∆t) → −∞, BEM’s amplification factor tends to 0, while TM’s tends to −1.Source: 2) Stiff decay property · p. 4

3) Hyperstability issue of BEM:

  • Hyperstability occurs when numerical integration produces a stable response for an unstable system; BEM cannot diagnose oscillatory instability when λ∆t satisfies the stated constraints.The cited BEM absolute-stability region includes the right half-plane outside the unit circle.Source: 3) Hyperstability issue of BEM · pp. 4–5
  • With ∆t = 1 s versus TM’s ∆t = 0.001 s, BEM matched the stable case’s final result but converged to the unstable equilibrium in the unstable case.The stable trajectory was coarse; the unstable case used a negative damping factor.Source: 3) Hyperstability issue of BEM · p. 5, Figure 2 · p. 5

B. Addressing Hyperstabibility Problem of BEM using Predictor-Corrector Approach

1) Cascading Failure Simulation Subprocesses (a):

  • The cascade subprocess uses variable-step BEM with OC, UVLS, and generator first-swing out-of-step relays, stopping when machine-speed variation is small with no anticipated relay actions or collapse occurs.Its stopping criterion is described as similar to TM’s.Source: 1) Cascading Failure Simulation… · p. 5
  • After each event, simulation uses ∆t = 0.002 s for k steps; Newton convergence requiring more than r iterations also triggers a reduction to ∆t = 0.002 s.The post-event steps capture non-oscillatory instability.Source: 1) Cascading Failure Simulation… · p. 5
  • Each cascade tier marks a topology-changing event and records its corresponding state x and voltage V; the subprocess solves a sequence of initial-value problems serially.The event records are denoted [ti : xi, Vi].Source: 1) Cascading Failure Simulation… · p. 5
  • BEM captures local voltage, frequency, and non-oscillatory angle instability, but oscillatory angle instability can misdirect the cascade, motivating detection of its earliest tier and ground-truth protective actions.The oscillatory category includes local or interarea angle instability.Source: 1) Cascading Failure Simulation… · p. 5

2) Predictor Subprocess (b): … C. Discussion on Computational Efficiency of BEM-PC

BEM-PC combines variable-step BEM, instability prediction, protection correction, and repeated IVP restarts. Its efficiency rationale is larger time steps and parallelizable predictor work, especially for long cascading-failure simulations.

2) Predictor Subprocess (b):

  • Predictor subprocess (b) runs short-duration independent IVPs in parallel, stops each when machine-speed variation falls below a threshold, and omits relay actions during that period.Variable-step BEM then reaches the post-event equilibrium, including the unstable equilibrium, and supplies the system matrix for the linearized model.Source: 2) Predictor Subprocess (b) · p. 5

3) Corrector Subprocess (c):

  • Eigendecomposition of the system matrix detects oscillatory instability and identifies its earliest event time; right eigenvectors and speed modeshapes locate participating machines for protection selection.For large systems, selective unstable-eigenvalue and corresponding right-eigenvector calculations are suggested.Source: 2) Predictor Subprocess (b) · p. 5, 3) Corrector Subprocess (c) · p. 6
  • At the predicted instability time, the corrector schedules the pre-determined protective action after its designed delay, such as generator tripping or an SPS action.The participating machines are identified from speed modeshapes, assuming the typical electromechanical-mode case.Source: 3) Corrector Subprocess (c) · p. 6

4) Restart subprocess (a):

  • The restart subprocess re-initiates IVPs from the predicted instability state, applies protection after a pre-defined delay, and repeats the process until no instability is detected.The paper reports that Section V-C’s results support its efficiency arguments and that comparison with the partitioned approach leads to similar conclusions.Source: 4) Restart subprocess (a) · p. 6, C. Discussion on Computational… · p. 6

C. Discussion on Computational Efficiency of BEM-PC

  • Variable-step BEM allows subprocess (a) to use significantly larger time steps than LTE-based variable-step TM, enabling faster execution.The passage attributes this to BEM’s stiff-decay property and notes that CPU-time and time-step comparisons are presented in Section V-C.Source: C. Discussion on Computational… · p. 6
  • Predictor subprocess (b) is parallelizable; it computes post-event equilibria, derives the system matrix using sparse computation, and performs eigendecomposition.The system-matrix calculation is a by-product of BEM, and the matrix need not be calculated for remaining equilibria once an unstable mode is found.Source: C. Discussion on Computational… · p. 6
  • BEM-PC’s efficiency relies on only a relatively small fraction of cascade simulations causing oscillatory instability and requiring subprocess (a) to be rerun.Subprocess (c) also needs minimal computation because it uses a lookup table to enact SPS action.Source: C. Discussion on Computational… · p. 6
  • For longer cascading-failure simulations, BEM’s larger integration time steps than TM during most of the process can outweigh BEM-PC’s extra prediction and correction computations.The paper contrasts this with short-term transient-stability simulations, where the speed gain is expected to be limited.Source: C. Discussion on Computational… · p. 6

IV. MODELING AND ADAPTIVE COI-FRAME-BASED APPROACH · A. Component and Relay Action Models · B. Adaptive COI-frame-based Approach

The approach models generators, loads, and protective relays, then adapts island-level center-of-inertia frames when a disturbance splits an island. Its stiff-decay property also allows BEM-PC to use larger variable time steps than trapezoidal simulation during most of the simulation.

A. Component and Relay Action Models

  • The component models include fourth-order synchronous generators with first-order governors and static exciters, static constant-power loads, and governor-free synchronous-condenser loads.Synchronous condensers use generator-like models except that they have no governors.Source: A. Component and Relay Action Models · p. 7
  • UVLS relays monitor average bus voltage and shed a load fraction when voltage remains below threshold; overcurrent, out-of-step, and oscillatory-instability SPS actions are also modeled.The overcurrent relays use average line current; generator out-of-step relays trip machines with non-oscillatory instability, while a pre-designed SPS addresses oscillatory instability involving multiple machines.Source: A. Component and Relay Action Models · p. 7

IV. MODELING AND ADAPTIVE COI-FRAME-BASED APPROACH

  • BEM-PC solves the same DAE initial-value problems as trapezoidal simulation but can reach post-disturbance equilibrium faster using larger variable time steps; its stiff decay also suits longer-term simulation.The framework can incorporate mid- or long-term stability issues such as boiler dynamics and long-term frequency instability.Source: A. Component and Relay Action Models · p. 7

B. Adaptive COI-frame-based Approach

  • The COI frame rotates at the inertia-weighted average machine speed and expresses rotor angles relative to the COI angle, supporting efficient Newton iteration under off-nominal frequency.The approach projects island phasors into the COI frame rather than the synchronous-speed network reference frame.Source: B. Adaptive COI-frame-based Approach · p. 7
  • When a parent island splits, its converged states cannot be reused directly because center-of-inertia frames apply locally; the adaptive approach transfers states into each child’s frame.The method addresses the mismatch between a parent’s frame and the locally applicable child-island frames.Source: B. Adaptive COI-frame-based Approach · p. 7
  • The islanding procedure maps states from the parent frame, updates them in each child’s COI frame, recomputes voltages, and uses the updated states and voltages to initialize the next step.After island formation, the standard COI-based approach applies until that island splits again.Source: B. Adaptive COI-frame-based Approach · pp. 7–8

V. CASE STUDIES

The case studies compare BEM-PC with TM on the IEEE 118-bus and Polish systems using random-outage cascades, with the IEEE 68-bus system introduced later. Predictor subprocesses are run serially, so reported speedups are conservative.

V. CASE STUDIES

  • Across 500 Monte Carlo runs per system, random initial outages trigger cascades in the 118-bus IEEE system and 2,383-bus Polish network.The IEEE 118-bus cases use 2 initial node outages and the Polish cases use 3; the models use synthetically generated dynamic data.Source: §V · p. 8
  • BEM-PC uses ∆tmin = 0.02 s, ∆tmax = 0.4 s, k = 6, r = 7, ϵ = 10^-4, and a maximum of 10 Newton iterations.The study also specifies relay settings, including T UV LS tp = 3 s, λ = 25%, and system-specific voltage thresholds.Source: §V · p. 8
  • Predictor subprocesses run serially, making the reported speedup a conservative estimate of the potential with parallelization.IEEE 118-bus simulations used an AMD Ryzen 7 3800X with 32 GB RAM; Polish simulations used four servers at PSU’s ROAR facility.Source: §V · p. 8

A. Monte-Carlo Simulation · 1) IEEE 118-Bus System:

In IEEE 118-bus Monte-Carlo simulations, BEM-PC closely matched TM’s cascade outcomes, showed high cascade-path agreement, and ran approximately 10 times faster on average.

1) IEEE 118-Bus System:

  • BEM-PC closely matched TM’s end-of-cascade demand-loss distribution, with small differences for 15%−25% demand loss.The comparison includes initial node outages.Source: 1) IEEE 118-Bus System · p. 8, Figure 6 · p. 8
  • BEM-PC closely matched TM’s end-of-cascade line-outage distribution, with small differences for 44−60 outages.The comparison includes initial node outages.Source: 1) IEEE 118-Bus System · p. 8, Figure 7 · p. 8
  • At cascade end, BEM-PC’s bus, machine, and line connection-state errors were small fractions of their respective totals.Although outliers increased average errors, BEM-PC replicated TM’s exact end result in almost all cases.Source: 1) IEEE 118-Bus System · p. 8, Table I · p. 9
  • BEM-PC’s central tendency measures of maximum voltage-magnitude, angle, and frequency errors were very small.The passage does not provide the individual metric values.Source: 1) IEEE 118-Bus System · p. 8, Table I · p. 9
  • The path-agreement measure R compares dependent branch outages under the same initial contingencies, with R = 1 indicating complete cascade-path agreement.For each contingency, the measure compares the sets of dependent line outages produced by the two models.Source: 1) IEEE 118-Bus System · pp. 8–9
  • BEM-PC showed high cascade-path agreement with TM and averaged approximately 10 times faster.The standard deviation of the path-agreement measure R was 0.05916.Source: 1) IEEE 118-Bus System · p. 9, Table I · p. 9

2) Polish System: · B. Hyperstability – A Challenge for BEM and Performance of Predictor-Corrector Approach

For the Polish system, BEM-PC closely reproduced TM’s cascade outcomes while substantially reducing runtime. In oscillatory-instability cases, its predictor-corrector approach addressed BEM hyperstability and reproduced ground-truth cascade behavior.

2) Polish System:

  • 34.6 times faster on average, BEM-PC closely matched TM on end-of-cascade demand loss and line outages; Table II reports an R standard deviation of 0.03230.The reported error measures indicate accurate end results and strong cascade-path agreement.Source: Figure 8 · p. 9, Figure 9 · p. 9, 2) Polish System · p. 9
  • Among 500 Polish-system runs with 3 initial node outages, 41 were resilient and 118 failed to converge; both groups were excluded from Figure 10 comparisons.The comparisons cover demand loss versus outages and cascade sequence time.Source: 2) Polish System · p. 9, Figure 10 · p. 10
  • For the remaining Polish cases, BEM-PC’s correlations and density distributions closely matched TM’s, although a few individual cases differed slightly.Figure 10 compares demand loss with outage counts and cascade sequence time.Source: 2) Polish System · p. 9, Figure 10 · p. 10
  • In a Polish sample case, BEM-PC followed TM’s exact cascade path and ran 28 times faster, with cascade sequence time exceeding 100 s in many cases.The sample timeline tracks branch outages and demand loss; the initial outages disconnected 10 lines at t = 3 s and caused significant demand loss.Source: 2) Polish System · p. 9, Figure 11 · p. 10

B. Hyperstability – A Challenge for BEM and Performance of Predictor-Corrector Approach

  • BEM-PC detects the unstable mode and participating machines by eigendecomposing the post-event equilibrium A matrix, enabling SPS protection actions.TM applies the SPS action after 7.5 s in IEEE 118-bus and 4.5 s in the Polish system; other actions, such as tripping lines, are possible.Source: B. Hyperstability – A Challenge for… · p. 10
  • In IEEE 118-bus case #1, plain BEM produced only one cascade tier, whereas BEM-PC detected oscillatory instability and, like TM, led to 25 tiers.BEM-PC estimated unstable modes 0.1239 ± j2.4889 and 0.1121±j2.0531, then tripped G51 and G39 after a 7.5 s delay.Source: B. Hyperstability – A Challenge for… · p. 10, Figure 12 · p. 10
  • Across oscillatory-instability cases #1 and #2, BEM-PC resolved BEM hyperstability, matched ground-truth cascade paths and end results, while BEM without correction differed significantly.Tables III and IV compare path agreement and end-of-cascade measures for the three models.Source: B. Hyperstability – A Challenge for… · p. 10

2) Polish System: · 3) IEEE 68-Bus NE-NY System:

On the Polish system, BEM-PC captures oscillatory instability and matches TM’s cascade endpoint and propagation path; on the IEEE 68-bus NE-NY system, it closely matches TM and achieves approximately 20× average speedup despite naturally occurring hyperstability.

2) Polish System:

  • BEM without predictor-corrector fails to capture Polish-system oscillatory instability, whereas BEM-PC estimates unstable modes and triggers SPS tripping of G286 and G287 after 4.5 s.For case #2, the estimated modes are 0.4306 ± j9.7845 and 0.4464 ± j9.3910.Source: 2) Polish System · pp. 10–11, Figure 14 · p. 11
  • BEM-PC matches TM’s cascade endpoints and propagation paths in both Polish-system cases, although two closely timed line trips occur in one tier for BEM-PC and two tiers for TM.For case #2, BEM without predictor-corrector stops after four tiers around 43 s.Source: Table VI · p. 11, 2) Polish System · p. 11
  • The IEEE 68-bus study targets a system that naturally exhibits oscillatory instability during cascade propagation, unlike the IEEE 118-bus and Polish systems.Source: 2) Polish System · p. 11

3) IEEE 68-Bus NE-NY System:

  • The system has 5 areas, 87 lines, and 16 generators; its least-damped mode is −0.0804 ± 2.4474j, and the study runs 500 Monte Carlo simulations with two random initial line outages.Simulations ran on an AMD Ryzen 7 3800X CPU with 32 GB RAM.Source: 3) IEEE 68-Bus NE-NY System · p. 11
  • The IEEE 68-bus simulations use T_OC = 4 s because the system exhibits low-frequency interarea modes.Source: 3) IEEE 68-Bus NE-NY System · p. 11
  • BEM-PC encounters hyperstability in 16.4% of IEEE 68-bus cases and corrects a predicted unstable mode by tripping line 41−42 after 5 s under the predefined SPS action.The unstable mode is 0.0061±2.3740j, with generators G14−G16 oscillating against the rest.Source: 3) IEEE 68-Bus NE-NY System · p. 11
  • On the IEEE 68-bus system, BEM-PC closely matches TM’s end-of-cascade demand-loss and line-outage comparisons while achieving approximately 20× average speedup.The reported comparison also includes end-state errors and path agreement, despite naturally occurring hyperstability.Source: 3) IEEE 68-Bus NE-NY System · p. 11

C. Analysis of Computational Efficiency · 1) Performance comparison with TM:

The analysis compares BEM-PC with TM and the partitioned approach, then examines how BEM-PC subprocesses contribute to runtime, including across 500 Polish-system cases.

C. Analysis of Computational Efficiency:

1) Performance comparison with TM:

2) Performance comparison with partitioned approach: … VI. CONCLUSION AND FUTURE WORK

BEM-PC addresses BEM’s hyperstability with a predictor-corrector using eigendecomposition at the post-event unstable equilibrium. It achieves substantial speedups over TM while following its cascade paths and end results in most cases.

2) Performance comparison with partitioned approach:

  • In the Polish grid, RK was slower than TM, and BEM-PC was faster than both: mean runtime ratios versus BEM-PC were 68.82 for RK and 34.54 for TM.For the 393 cases simulated with RK, median ratios were 51.84 for RK and 24.44 for TM; the other 107 cases exceeded 60 hours. In IEEE 118-bus, RK was slightly faster than TM, but the mean ratios were 7.72 for RK and 9.96 for TM.Source: 2) Performance comparison with… · pp. 13–14

D. Comparison with AC-QSS and classical models

  • In IEEE 118-bus cascades, AC-QSS and classical models resembled the 4th-order ground truth for less severe cascades, then diverged as outages and load tripping increased; AC-QSS was optimistic and classical pessimistic.Ground truth used a 4th-order generator model solved with TM.Source: D. Comparison with AC-QSS and… · p. 14, Figure 22 · p. 14
  • The most poorly damped modes differed between 4th-order and classical NE-NY models: −0.0804 ± 2.4474j versus −0.0718 ± 5.0125j, with different generator groups oscillating.In the 4th-order model, G14–16 oscillated against NETS and NYPS; in the classical model, G15 oscillated against G14 and G16.Source: D. Comparison with AC-QSS and… · p. 14, Figure 23 · p. 14
  • For NE-NY cases with nonzero errors, the classical model’s mean demand-loss error exceeded 40% and its mean line-outage error exceeded 40; AC-QSS errors were similar but occurred in more cases.Table XI counts cases with nonzero errors, and Figure 24 quantifies error among those cases.Source: D. Comparison with AC-QSS and… · pp. 14–15, Figure 24 · p. 14, Table XI · p. 14

3) Polish System:

VI. CONCLUSION AND FUTURE WORK

  • BEM-PC achieved average speedups of ≈10×, ≈20×, and ≈35× over TM in the IEEE 118-bus, IEEE 68-bus, and Polish systems, respectively, while matching TM cascade paths and end results in most cases.The comparisons used 500 random-node-outage simulations per system; BEM-PC also retained a computational advantage over the partitioned approach.Source: §VI · p. 15
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