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Dynamic Modeling of Cascading Failure in Power Systems

Jiajia Song, Eduardo Cotilla-Sanchez, Goodarz Ghanavati, Paul D. H. Hines

arXiv:1411.3990v2physics.soc-pheess.SY

TL;DR

Cascading-failure modeling must account for interacting mechanisms that existing models often omit. The paper introduces COSMIC, evaluates it across contingencies and load configurations, and compares it with a dc QSS model.

  • Problem

    Concurrent modeling of power-system dynamics and discrete protection events is challenging and omitted from most existing cascading-failure models.

  • Method

    COSMIC uses hybrid discrete/continuous differential-algebraic equations to simulate nonlinear power-system dynamics, protection systems, load responses, and cascading events.

  • Results

    COSMIC produces heavy-tailed blackout-size distributions, load configurations yield 2.5%–3.5% probabilities of large demand losses, and dynamic and dc QSS models differ substantially later in cascades.

  • Takeaways & Limitations

    Detailed dynamic models help assess the relative importance of cascading mechanisms and modeling assumptions, while load models materially affect simulated cascading risk.

  • Takeaways & Limitations

    COSMIC is likely too slow for many large-scale statistical analyses, and the observed statistics may be affected by the limited contingency sample.

Abstract

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The modeling of cascading failure in power systems is difficult because of the many different mechanisms involved; no single model captures all of these mechanisms. Understanding the relative importance of these different mechanisms is an important step in choosing which mechanisms need to be modeled for particular types of cascading failure analysis. This work presents a dynamic simulation model of both power networks and protection systems, which can simulate a wider variety of cascading outage mechanisms, relative to existing quasi-steady state (QSS) models. The model allows one to test the impact of different load models and protections on cascading outage sizes. This paper describes each module of the developed dynamic model and demonstrates how different mechanisms interact. In order to test the model we simulated a batch of randomly selected $N-2$ contingencies for several different static load configurations, and found that the distribution of blackout sizes and event lengths from the proposed dynamic simulator correlates well with historical trends. The results also show that load models have significant impacts on the cascading risks. This dynamic model was also compared against a QSS model based on the dc power flow approximations; we find that the two models largely agree, but produce substantially different results for later stages of cascading.

I. INTRODUCTION

COSMIC addresses the difficulty of modeling diverse cascading-outage mechanisms by combining nonlinear network dynamics, load responses, and discrete protection events in a hybrid simulation framework.

  • COSMIC is a nonlinear dynamic model designed to study a wide variety of cascading-outage mechanisms.
  • Existing QSS models can represent cascading overloads but generally omit nonlinear mechanisms such as voltage collapse and dynamic instability.
  • COSMIC combines differential equations for machines and other dynamic components with nonlinear power-flow equations and explicitly modeled load-voltage responses.
  • Discrete equations represent component failures and load shedding by monitoring proximity to thresholds that trigger changes.
  • COSMIC provides an open, extensible platform with adaptive time steps, recursive islanded simulations, and HPC integration for testing modeling assumptions.
  • The model supports configurable load models, including constant P, I, Z, and exponential forms, because load assumptions can strongly affect dynamics and convergence.

B. Relay modeling

COSMIC models protective actions as delayed discrete events driven by electrical and thermal thresholds, covering line protection and load-shedding functions.

  • Protective-relay time delays prevent brief transient oscillations from immediately triggering relay actions.
  • COSMIC implements fixed-time and time-inverse triggering through counters that respond to threshold violations and recover when signals return within limits.
  • Five relay types are modeled: over-current, distance, temperature, under-voltage load shedding, and under-frequency load shedding.
  • Temperature relays use line-current-driven thermal dynamics; parameters make rate-A current reach 75 °C and a rate-A-to-rate-C jump trigger in 60 seconds.
  • UVLS and UFLS shed 25% of initial Pd,i by default after voltage or frequency thresholds are violated, using fixed delays of 0.5 seconds.

C. Solving the hybrid DAE

COSMIC solves its hybrid DAE by integrating continuous dynamics adaptively, locating discrete events, updating the network equations, and recursively handling island formation.

  • The trapezoidal rule simultaneously integrates and solves COSMIC’s differential and algebraic equations.
  • Variable time steps become smaller during high-deviation transitions and larger as oscillations damp toward steady-state values.
  • Adaptive stepping retains the event time through linear interpolation between time steps.
  • At a discrete event, the simulator stops the previous DAE solve, processes the event, and resumes with updated algebraic variables while holding differential variables.
  • When an event changes network connectivity, COSMIC inspects Ybus, separates the network into islands, and recursively simulates the resulting subnetworks.

D. Validation

The validation studies compare COSMIC with established simulation software and organize time-domain computation around event processing, network separation, and updated DAE solutions.

  • COSMIC’s dynamic response matched PowerWorld within 0.11% MAE on random contingencies in the classic IEEE 9-bus test case.
  • Time-Domain Simulation Algorithm: The time-domain algorithm initializes a hybrid DAE system, processes exogenous contingencies, and checks updated Ybus for network separation.
  • Time-Domain Simulation Algorithm: Separated networks are divided into subnetworks and simulated recursively from the initialization step.
  • Time-Domain Simulation Algorithm: After a configuration change, COSMIC recomputes Ybus and resolves the algebraic system before integrating until the end time or a discrete threshold crossing.
  • Time-Domain Simulation Algorithm: At detected events, the algorithm interpolates differential and algebraic variables and changes relay states according to the event constraints.
  • Validation caveat: Agreement with PowerWorld required setting several simplified-model time constants to zero or nearly zero.

III. EXPERIMENTS AND RESULTS

Experiments evaluate COSMIC’s computational formulations, relay-event handling, and cascading-event behavior across test systems. Results show formulation differences depend on system scale, while relay timing can produce islanding and prevent subsequent load shedding.

  • Experimental design: Experiments used the 9-bus, 39-bus, and 2383-bus systems to examine formulation efficiency, relay functions, cascading events, load modeling, and comparisons with a dc-power-flow model.The 2383-bus system represents an equivalenced Polish-network snapshot.
  • Computational efficiency: For the 39-bus case, the rectangular formulation required fewer linear solves than the polar formulation across different demand-loss groups.The comparison used 45 N−1 and 222 randomly selected N−2 experiments, each ending at 50 seconds with the same demand loss across formulations.
  • Computational efficiency: For the 2383-bus case, rectangular and polar formulations produced almost identical numbers of linear solves, with no significant rectangular-formulation improvement.The experiments included 2494 N−1 and 556 N−2 contingencies.
  • Computational efficiency: The 2383-bus simulations required fewer linear solves than the 39-bus simulations, suggesting greater branch-outage impact and more dynamic oscillations in the smaller network.This interpretation is presented as a suggestion rather than a general conclusion.
  • Relay event illustration: In the 9-bus relay example, a distance-relay trip 0.5 seconds after the initial outage split the system into two isolated islands.The initial outage was the Bus 6–Bus 9 line at t = 10 seconds, and the DIST relay acted on branch 5-7.
  • Relay event illustration: UVLS did not trigger because load shedding restored the voltage trace to its threshold before the UVLS timer expired, resetting its 0.5-second delay.The dashed black line in Fig. 2 marks the UVLS voltage threshold.

C. Cascading outage examples using the 39-bus and the 2383-bus power systems

The examples show how COSMIC represents cascading outages as time-ordered interactions among dynamic responses, protection actions, islands, and load shedding. Across randomized N−2 simulations, load composition substantially affects blackout risk, while event counts and durations correlate only weakly with blackout size.

  • 39-bus example: In the 39-bus example, dynamic oscillation after two initial outages led to relay trips, load shedding, islanding, and a declared blackout area.The first over-current relay triggered after approximately 55 seconds; later outages produced three isolated networks, one algebraically unsolvable because of power imbalance.
  • 2383-bus example: The 2383-bus example began with two exogenous outages and produced 92 discrete events, including 24 branch outages and a small island.Load shedding occurred at buses marked with red squares, and affected low-voltage or low-frequency buses could be far from the initiating outages.
  • 2383-bus example: The cascade initially progressed slowly but accelerated as failures accumulated, ending in rapid collapse with most branch outages and load-shedding events.The timeline separates branch outage events from the associated load-shedding events.
  • N−2 contingency analysis: Across four 1200-contingency groups, constant Z loads had the lowest average loss and large-blackout risk, while constant E loads had the highest risk of losses above 1000 MW.Constant P loads followed a trend similar to constant E loads, and the equal four-way mixture performed between constant P and constant Z loads.
  • N−2 contingency analysis: 2.5% to 3.5% were the reported probabilities of large demand losses across the four load configurations.The study states that load models significantly affect dynamic simulation and may increase non-convergence when modeled improperly.
  • N−2 contingency analysis: The number of events and cascade duration each had a positive but weak correlation with blackout size measured in MW.Event-length distributions for constant P and constant E loads had comparable patterns and generally fewer equivalent branch outages than the other configurations.

E. Comparison with a dc cascading outage simulator

The paper compares COSMIC with a dc QSS cascading-outage simulator because dynamic models represent mechanisms absent from dc approximations. The comparison examines both demand-loss probabilities and agreement in cascade patterns to identify where the simpler model is informative and where dynamic modeling matters.

  • Model comparison: Dc QSS models are numerically stable and can reproduce statistically similar results to real-system data, but they simplify the system substantially.They can describe cascading overloads but do not capture nonlinear mechanisms such as voltage collapse or dynamic instability.
  • Model comparison: Dynamic models include cascading mechanisms that cannot be represented in the dc model.The comparison therefore evaluates overall demand-loss probabilities and agreement between cascade patterns.
  • Model comparison: The two experiments are designed to identify dominant cascading mechanisms at different stages and conditions favoring dynamic rather than simpler modeling.The supplied passages describe the comparison objectives but do not report its numerical outcomes.

1) The probabilities of demand losses:

COSMIC and the dc simulator produce different demand-loss distributions, with COSMIC showing much larger losses in some cases. Numerical non-convergence, protection differences, sampling, and model assumptions help explain the divergence.

  • Demand-loss probabilities: 24602 MW versus 2639 MW: COSMIC produced a much larger maximum demand loss than the dc simulator, with probabilities of 2.5% versus 0.08%.The comparison used the same 1200 randomly selected N −2 branch outage pairs and a constant impedance load in COSMIC.
  • Demand-loss probabilities: COSMIC’s numerical failures contributed substantially to larger blackout sizes because failed DAE solutions were treated as complete network or sub-network blackouts.This reflects a tradeoff between more detailed component models and assumptions that can affect outcome accuracy.
  • Sampling limitation: 1200 contingencies represented 0.0278% of all N −2 branch outage pairs, so the observed statistics may depend on the sample size and sampling approach.The paper leaves broader investigation of sampling effects for future work.
  • Path agreement: 0.1948: Average path agreement across 336 critical outage pairs was low, indicating substantial differences between COSMIC and dc cascade paths.The dc model’s tendency to produce longer cascades increases the denominator in the path-agreement measure.
  • Path agreement: 0.3487: Average path agreement increased when only the first 10 branch outages were compared, with some paths matching perfectly at R = 1.The models tend to agree during early cascading stages and diverge later, when nonlinear dynamics become more pronounced.
  • Implications: Detailed dynamic models can clarify the relative importance of cascading mechanisms, but COSMIC may be too slow for many large-scale statistical analyses.Comparisons with simpler models help assess the importance of modeling assumptions.

V. APPENDIX

COSMIC represents generator behavior with differential equations for machine dynamics and nonlinear equations for power outputs. The adapted salient-pole formulation forms a third-order generation-machine model.

  • Machine dynamics: COSMIC uses a third-order machine model based on polar-formulation differential equations for generator dynamics.The model combines the swing equation, rotor-angle equation, and transient-voltage dynamics.
  • Machine dynamics: The rotor-speed equation describes synchronous-machine acceleration or deceleration from mechanical input, electrical output, inertia, and damping.Normalized rotor speed remains fixed during normal operation and changes after disturbances.
  • Machine dynamics: The rotor angle δ_i is obtained by integrating relative rotor-speed change with respect to synchronous speed.This links speed deviations to the machine’s angular state.
  • Generator electrical model: The adapted salient-pole model uses nonlinear equations for active and reactive generator power outputs.Transient open-circuit voltage magnitude is governed by a differential equation involving the direct-axis transient time constant and exciter output.

4) Exciter:

COSMIC models excitation with a generic second-order differential system. The exciter responds to reference and terminal voltages while limiting output between specified bounds.

  • Exciter: COSMIC’s generic exciter is a second-order differential system for machine excitation control.The configuration is illustrated in Fig. 10.
  • Exciter: The exciter uses reference voltage Vref and terminal voltage Vt to determine excitation behavior.TA, TB, and KE are exciter time constants.
  • Exciter: A differentiable sigmoidal function limits exciter output between Emin and Emax.This limiter is part of the generic exciter formulation.

5) Governor:

COSMIC uses a governor model combining droop and approximate integral control, then advances its continuous states with a trapezoidal numerical method. Solver and relay parameters are specified separately.

  • Governor: COSMIC’s governor combines proportional droop control with approximate secondary integral control to restore frequency deviations to zero.The model includes droop, PI, and servo-motor time constants, with P3 as an intermediate PI-controller variable.
  • Numerical solution: The trapezoidal solution method computes next-step continuous states x+ and y+ from the current variables and differential-algebraic functions.It solves a nonlinear system over the timestep Δt.
  • Model settings: Numerical-solver parameters and protective-relay settings are listed separately for COSMIC.These settings are presented in Tables VI and VII.
  • Numerical solution: The numerical timestep is based on 1/2 · Δt_prev except when a discrete event occurs.Discrete events therefore alter the ordinary timestep rule.
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