[Paper Review] Approaching the Transient Stability Boundary of a Power System: Theory and Applications

Peng Yang, Feng Liu+2 more|arXiv (Cornell University)|Sep 26, 2021

Power System Optimization and Stability4 citations

TL;DR

This paper proposes a generalized expansion methodology to improve transient stability boundary estimation in power systems by relaxing the requirement for an initial guess to be a strict subset of the stability region. Using flow mapping and diffeomorphism theory, it enables local and global convergence toward the true stability boundary, significantly enhancing accuracy of critical clearing time (CCT) estimates with minimal computational overhead, as validated on the IEEE 39-bus system.

ABSTRACT

Estimating the stability boundary is a fundamental and challenging problem in transient stability studies. It is known that a proper level set of a Lyapunov function or an energy function can provide an inner approximation of the stability boundary, and the estimation can be expanded by trajectory reversing methods. In this paper, we streamline the theoretical foundation of the expansion methodology, and generalize it by relaxing the request that the initial guess should be a subset of the stability region. We investigate topological characteristics of the expanded boundary, showing how an initial guess can approach the exact stability boundary locally or globally. We apply the theory to transient stability assessment, and propose expansion algorithms to improve the well-known Potential Energy Boundary Surface (PEBS) and Boundary of stability region based Controlling Unstable equilibrium point (BCU) methods. Case studies on the IEEE 39-bus system well verify our results and demonstrate that estimations of the stability boundary and the critical clearing time can be significantly improved with modest computational cost.

Motivation & Objective

• To address the limitation of existing direct methods in transient stability analysis, which produce overly conservative stability boundary estimates.
• To generalize the trajectory reversing method by removing the strict requirement that the initial guess must be a subset of the stability region.
• To develop theoretically grounded, computationally efficient algorithms that improve boundary and critical clearing time (CCT) estimation for both lossless and lossy power system models.
• To validate the effectiveness of the proposed expansion methodology on practical power system networks, particularly the IEEE 39-bus system.
• To demonstrate robustness in the presence of model inaccuracies, such as those arising from lossy system representations.

Proposed method

• The paper establishes a theoretical foundation using flow mapping and diffeomorphism to characterize the evolution of level sets under reverse integration, enabling convergence to the true stability boundary.
• It generalizes the trajectory reversing method (TRM) to allow initial guesses that are only partially within the stability region, broadening applicability to local methods like PEBS and BCU.
• The proposed expansion algorithms iteratively refine the stability boundary estimate by applying reverse integration along trajectories, improving the level set approximation.
• The method leverages the monotonicity of trajectories and the structure of energy functions to ensure convergence toward the exact boundary.
• For CCT estimation, the algorithm is integrated with the BCU method, using iterative expansion to refine critical clearing time predictions.
• The approach is implemented using numerical integration with adaptive step size, minimizing computational cost while maximizing accuracy gains.

Experimental results

Research questions

• RQ1 Can the trajectory reversing method be generalized to work with initial guesses that are not fully contained within the stability region?
• RQ2 How can the theoretical foundation of boundary expansion be strengthened using flow mapping and diffeomorphism in nonlinear dynamical systems?
• RQ3 To what extent can the proposed expansion algorithms improve the accuracy of stability boundary and critical clearing time (CCT) estimates in power systems?
• RQ4 How does the method perform under model inaccuracies, such as in lossy system representations where energy functions are approximate?
• RQ5 What is the trade-off between estimation accuracy and computational cost when applying the expansion algorithm in practical power system applications?

Key findings

• The proposed expansion algorithms monotonically improve the accuracy of stability boundary estimation in the lossless IEEE 39-bus system, validating the theoretical convergence properties.
• In the lossy case, despite fluctuations due to energy function inaccuracies, the algorithms still converge to more accurate boundary estimates, demonstrating robustness.
• On average, six expansion iterations reduced the CCT estimation error by 10.69% while increasing computation time by only 0.07 seconds—equivalent to 16.18% of the original BCU method’s time cost.
• For Fault No.1, the algorithm achieved a 10.45% reduction in CCT estimation error after six expansions, with minimal time overhead.
• The method significantly outperforms standard PEBS and BCU methods in terms of accuracy, especially in reducing conservativeness without requiring full knowledge of the stability region.
• The expansion methodology remains effective even with inaccurate energy functions, suggesting potential for application in systems where exact Lyapunov functions are unavailable.