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Lyapunov Functions Family Approach to Transient Stability Assessment

Thanh Long Vu, Konstantin Turitsyn

arXiv:1409.1889v3eess.SY

TL;DR

Transient-stability assessment is difficult because post-fault power-system dynamics are strongly nonlinear and direct simulation wastes resources on safe contingencies. The paper constructs a Lyapunov Functions Family with Semi Definite Programming, adapts certificates to initial conditions, and reports broader certification than the closest-UEP energy method with tractable convex computations.

  • Problem

    Transient-stability assessment for strongly nonlinear post-fault dynamics is computationally challenging, while direct simulation expends resources on contingencies that are safe.

  • Method

    The paper constructs a convex family of Lyapunov functions through Semi Definite Programming and adapts function selection to specific contingency initial conditions.

  • Results

    The LFF approach generally certifies broader, less conservative stability regions than the closest-UEP energy method and remains tractable for large-scale systems.

  • Takeaways & Limitations

    A family of certificates enables contingency-specific stability assessment, including cases not certified by the closest-UEP energy method.

  • Takeaways & Limitations

    The approach requires further improvement for realistic generator, load, and transmission models before industrial deployment.

Abstract

from arXiv · show

Analysis of transient stability of strongly nonlinear post-fault dynamics is one of the most computationally challenging parts of Dynamic Security Assessment. This paper proposes a novel approach for assessment of transient stability of the system. The approach generalizes the idea of energy methods, and extends the concept of energy function to a more general Lyapunov Functions Family (LFF) constructed via Semi-Definite-Programming techniques. Unlike the traditional energy function and its variations, the constructed Lyapunov functions are proven to be decreasing only in a finite neighborhood of the equilibrium point. However, we show that they can still certify stability of a broader set of initial conditions in comparison to the traditional energy function in the closest-UEP method. Moreover, the certificates of stability can be constructed via a sequence of convex optimization problems that are tractable even for large scale systems. We also propose specific algorithms for adaptation of the Lyapunov functions to specific initial conditions and demonstrate the effectiveness of the approach on a number of IEEE test cases.

I. INTRODUCTION

The paper addresses inefficient transient-stability assessment by extending classical energy methods to a family of Lyapunov functions constructed with convex optimization. The family supports contingency-specific certification and can provide broader stability regions than the closest-UEP method.

  • Motivation: Direct simulations accurately assess large systems but waste computational resources because most contingencies are safe.The paper motivates mathematically structured screening methods as alternatives for non-critical scenarios.
  • Prior approaches: Classical energy methods rigorously certify safety and enable fast contingency screening through Lyapunov-based analysis.Energy functions are presented as a specific form of Lyapunov functions guaranteeing convergence to stable equilibria.
  • Contributions: The proposed Lyapunov Functions Family contains a convex set of functions, each yielding a different estimate of the stability region.This generalizes the classical energy-function approach rather than selecting a single fixed certificate.
  • Contributions: An adaptation algorithm selects a Lyapunov function suited to specific contingency situations or initial conditions.The approach evaluates functions at post-fault states and can iteratively seek a certificate for a given initial condition.
  • Contributions: The method offers broader stability certification than the closest-UEP method without requiring the fault-on trajectory used by the controlling-UEP method.Its Lyapunov-function family is constructed through a sequence of convex Semi Definite Programming problems.
  • Applications: Stability certification can trade conservatism against computation by using nonconvex boundary minimization or faster polynomial convex optimization.The nonconvex option can certify larger regions, while convex alternatives provide more conservative but faster certification.

II. TRANSIENT STABILITY OF POWER SYSTEMS

The paper formulates post-fault transient stability as convergence to a stable equilibrium and characterization of its region of attraction. It uses a state-space swing-equation model that separates nonlinear sinusoidal interactions into componentwise functions suitable for linear bounding.

  • Problem formulation: Faults and component disconnections drive the system away from its pre-fault equilibrium, creating transient post-fault dynamics after reclosing.The assessment problem concerns whether these dynamics converge to the post-fault equilibrium.
  • System model: The model uses swing equations with static-impedance loads, perfectly voltage-controlled generators, inertia, damping, network susceptance, and mechanical torque.Generator voltage magnitudes are assumed constant, while excitation, network losses, and dynamic load response are omitted.
  • Problem formulation: The study seeks the region of attraction: initial angle deviations and angular velocities from which the system converges to the equilibrium.The equilibrium is characterized by rotor-angle differences, despite nonuniqueness under uniform angle shifts.
  • State-space representation: A state-space vector combines rotor-angle deviations and angular velocities, enabling nonlinear-control techniques to be applied to the transient dynamics.The matrix representation separates linear dynamics from the nonlinear transformation of angle differences.
  • Nonlinear structure: The nonlinear transformation is componentwise and trigonometric, with each edge contributing a sinusoidal interaction between generator-angle differences.This diagonal structure permits bounds on the nonlinear terms by linear functions.
  • Nonlinear structure: Within the polytope defined by |δ_kj + δ*_kj| < π, the nonlinear interactions can be bounded, while monotonicity holds in the smaller region |δ_kj| < π/2.These properties support the construction of Lyapunov functions and convexity arguments for their level sets.

III. FAMILY OF LYAPUNOV FUNCTIONS FOR STABILITY ASSESSMENT

The paper generalizes energy methods by constructing a convex family of Lyapunov functions whose decay and certified stability regions are local but adaptable. Different constructions trade certification conservatism against computational tractability.

  • Classical energy functions combine turbine kinetic energy with network potential energy and decrease because of damping.
  • For stable linear systems, nonnegative combinations of modal Lyapunov functions form a convex cone, motivating a Lyapunov Functions Family for nonlinear systems.
  • The proposed family is defined through linear matrix inequalities, with each feasible parameter choice producing a Lyapunov function generalizing kinetic and potential energy terms.
  • Each Lyapunov function strictly decreases inside a phase-space polytope, where trajectories are guaranteed to converge to the normal equilibrium point.
  • Invariant stability estimates are formed by combining Lyapunov sublevel sets with the polytope and excluding flow-out boundary escape.
  • Different family members yield different stability estimates, enabling adaptation and unions of certified sets; convex formulations are faster but more conservative, while nonconvex optimization gives larger estimates.

IV. DIRECT METHOD FOR CONTINGENCY SCREENING

The direct screening method evaluates candidate Lyapunov functions at post-fault states and compares those values with stability thresholds. An iterative convex procedure adapts the function to the initial condition, while threshold constructions trade conservatism for computational cost.

  • A post-fault state is certified when its Lyapunov value V0 is below the corresponding threshold Vmin.
  • The least-conservative threshold uses nonconvex optimization, the convex formulation is faster but more conservative, and the analytical approximation requires no optimization but gives conservative guarantees.
  • The Lyapunov Functions Family provides a cone of choices that can be adapted to a specific initial condition or family of conditions.
  • The adaptation algorithm repeatedly solves the LMI problem while imposing a stricter value at the initial condition, preserving convexity because that value is linear in the decision matrices.
  • The iteration terminates after finitely many steps when a Lyapunov function certifies the initial condition, if such a certificate exists.
  • In the illustrated 2-bus case, successive iterations produced increasingly large stability regions until one contained the target initial state.

A. Classical 2 bus system

The classical 2-bus example compares convex and non-convex LFF certificates with the closest-UEP energy method. LFF identifies stable initial conditions missed by the energy method, while adaptation enlarges certified regions until they contain the target state.

  • A. Classical 2 bus system: The 2-bus system provides a visual state-space test of LFF effectiveness using a single second-order differential equation.The example compares convex and non-convex Lyapunov-function invariant sets with the closest-UEP energy stability region.
  • A. Classical 2 bus system: LFF certifies many contingency states that the closest-UEP energy method cannot certify.The comparison is made for initial configurations x0 in the illustrated state space.
  • A. Classical 2 bus system: The non-convex Lyapunov function gives a less conservative certificate than the convex function, but requires additional computational overhead.The convex alternative is more conservative but computationally cheaper.
  • A. Classical 2 bus system: The adaptation algorithm produces increasingly large stability regions until one contains the specified initial state x0.This demonstrates condition-specific selection of a Lyapunov function from the family.

B. Kundur 9 bus 3 generator system

In the 9-bus, 3-generator case, the closest-UEP energy method fails to certify a specified contingency, whereas an adapted LFF certifies it and simulation confirms convergence to equilibrium.

  • B. Kundur 9 bus 3 generator system: The study evaluates post-fault dynamics of the Kundur 9-bus, 3-generator system using voltage, mechanical-input, and reduced-admittance data.The equilibrium and system parameters are derived from the tabulated model data.
  • B. Kundur 9 bus 3 generator system: The closest-UEP energy method cannot certify the contingency because E(0) = 0.4943 exceeds the critical energy of about 0.196.The initial state is {δ12(0) = 2.513, δ13(0) = 0.7854}.
  • B. Kundur 9 bus 3 generator system: An LFF certifies the same contingency through strict Lyapunov-function decrease, and simulation confirms convergence from the initial state to equilibrium.The result directly contrasts the energy-method certificate with the proposed Lyapunov-function certificate.

C. New England 39 bus 10 generator system

The New England 39-bus, 10-generator case tests whether LFF construction remains computationally tractable as system size increases.

  • C. New England 39 bus 10 generator system: The 39-bus, 10-generator experiment constructs a Lyapunov function by solving LMIs for a 20×20 matrix Q and 45×45 diagonal matrices K and H.The equilibrium is obtained from power-flow-like equations, and the LMIs are solved with MATLAB CVX.
  • C. New England 39 bus 10 generator system: The reported LMI solution takes about 2.5s on a normal laptop.This timing is presented as evidence for practical computational scalability in the tested system.

VI. DISCUSSION OF THE RESULTS

The discussion presents LFF as a generalization of the energy method that can yield less-conservative, adaptable certificates while retaining computational tractability. Its decay guarantee is limited to a finite polytope, requiring additional state-containment conditions and introducing a scalability trade-off.

  • VI. DISCUSSION OF THE RESULTS: LFF generalizes the classical energy method by providing multiple Lyapunov functions that decay near the equilibrium point.Different functions in the family yield different stability-region estimates.
  • VI. DISCUSSION OF THE RESULTS: The Lyapunov functions are certified to decay only within the finite polytope defined by bounded generator-angle differences.The stated condition is |δij + δ*ij| < π.
  • VI. DISCUSSION OF THE RESULTS: Exceedingly large angle differences can cause high line currents and activate protective relays absent from the swing-equation model.This explains the practical boundary associated with the finite-region decay guarantee.
  • VI. DISCUSSION OF THE RESULTS: Stability certification must ensure trajectories remain inside the polytope, creating a trade-off between least-conservative NP-hard inscription and polynomial conservative bounds.The convex and algebraic alternatives are intended to scale to large systems.
  • VI. DISCUSSION OF THE RESULTS: LFF certificates are generally less conservative than closest-UEP energy certificates and may remain tractable for large-scale systems.The experiments also report application to medium-sized models on regular laptops.

VII. PATH FORWARD

The paper identifies extensions needed before industrial deployment and outlines how Lyapunov-function certificates could support faster, broader security assessment. Key boundaries include model extensions, uncertainty robustness, and inability to assess unstable post-fault states.

  • VII. PATH FORWARD: Offline certificate databases could enable nearly instantaneous assessment for common contingencies while reserving direct simulations for uncertified events.The proposed extension could also certify regions of operating-condition space against common contingencies.
  • VII. PATH FORWARD: Industrial deployment requires extending the approach beyond the simplest transient-dynamics model to more realistic generators, loads, and transmission networks.The authors state that no technical barriers prevent generalization, because nonlinearity remains confined to individual components interacting through a linear network.
  • VII. PATH FORWARD: Network losses and first-order dynamic load models can be incorporated relatively easily, whereas higher-order generator and load models pose the main technical challenge.The difficult extension is establishing an analogue of the bound used by the algorithm for higher-order models.
  • VII. PATH FORWARD: Sum-Of-Squares polynomial algebraic geometry is identified as a potential tool for bounding complicated algebraic nonlinearities in higher-order models.The motivation is that individual generator models have relatively small order independent of system size.
  • VII. PATH FORWARD: Uncertainty in system parameters and initial states remains an important robustness question for the algorithm.Because the method bounds nonlinearity, certificates could extend to subsets of equilibria and initial post-fault states, though likely more conservatively.
  • VII. PATH FORWARD: The proposed algorithms cannot assess situations in which the post-fault state is unstable.Certifying transient instability is left as a possible direction for future research.

A. Proof of the Lyapunov function decay in the polytope P

The proof establishes that the Lyapunov function decreases throughout the polytope P and uses invariance and boundary arguments to show convergence from an appropriate invariant subset.

  • A. Proof of the Lyapunov function decay in the polytope P: The matrix identities from (8) express the relevant quadratic terms through matrices X and Y, enabling the derivative calculation for V(x).These identities relate ATQ + QA, QB − CT H − (KCA)T, and −2H to factorizations involving X and Y.
  • A. Proof of the Lyapunov function decay in the polytope P: The derivative expression combines the quadratic term, nonlinear interaction F, and the system derivative ẋ before applying structural identities.The intermediate expression is 0.5xT(ATQ + QA)x − xTQBF + FTKCẋ.
  • A. Proof of the Lyapunov function decay in the polytope P: Using CB = 0 and YT Y = 2H transforms the derivative into a form suitable for proving nonpositivity.The identities eliminate terms needed to complete the relevant quadratic structure.
  • A. Proof of the Lyapunov function decay in the polytope P: The resulting derivative is the sum of a negative quadratic term and a nonlinear term involving H, Xx − YF, and Cx − F.Specifically, V̇(x) = −0.5(Xx − YF)T(Xx − YF) − (Cx − F)T HF.
  • A. Proof of the Lyapunov function decay in the polytope P: The sector property g{k,j} ≥ 0 for admissible angle differences implies V̇(x) ≤ 0 throughout P, so V(x) is decaying there.The stated condition applies when |δkj + δ*kj| ≤ π.
  • A. Proof of the Lyapunov function decay in the polytope P: For an initial state in invariant set R, LaSalle’s principle yields convergence toward the set where V̇(x) = 0.The proof then uses the zero-derivative condition to constrain the angle differences and nonlinear term.
  • A. Proof of the Lyapunov function decay in the polytope P: The zero-derivative set forces F = 0, after which the dynamics reduce to ẋ = Ax and trajectories converge to stationary points.The trajectory therefore converges either to the stable equilibrium or to a stationary point on the boundary of P.
  • A. Proof of the Lyapunov function decay in the polytope P: Boundary convergence is ruled out because decay from V(x0) < Vmin contradicts the boundary value V(x*) ≥ Vmin.This contradiction establishes convergence to the stable equilibrium for the considered initial state.
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