Frequency-Aware Sparse Optimization for Diagnosing Grid Instabilities and Collapses

TL;DR

The paper tackles grid survivability under disturbances by integrating primary frequency response into a steady-state, circuit-theoretic power-flow framework. It introduces a frequency-aware sparse optimization that jointly minimizes corrective injections and enforces frequency stability via droop relationships $F_j(\\Delta f)$ and bounds $\Delta f_{ ext{min}}\le \Delta f\le \Delta f_{ ext{max}}$, yielding sparse pinpointing of critical compensations. Key findings include successful localization of dominant vulnerabilities (e.g., four buses in a 1354-bus case with a $3424.8$ MW outage) and detection of necessary slack actions to maintain $\,\Delta f$ within bounds, with runtimes under four minutes on a laptop for large-scale systems. This approach provides a scalable, actionable diagnostic tool for operators and planners to identify minimal, geographically localized remedies to preserve power balance and frequency stability after contingencies.

Abstract

This paper aims to proactively diagnose and manage frequency instability risks from a steady-state perspective, without the need for derivative-dependent transient modeling. Specifically, we jointly address two questions (Q1) Survivability: following a disturbance and the subsequent primary frequency response, can the system settle into a healthy steady state (feasible with an acceptable frequency deviation $Δf$)? (Q2) Dominant Vulnerability: if found unstable, what critical vulnerabilities create instability and/or full collapse? To address these questions, we first augment steady-state power flow states to include frequency-dependent governor relationships (i.e., governor power flow). Afterwards, we propose a frequency-aware sparse optimization that finds the minimal set of bus locations with measurable compensations (corrective actions) to enforce power balance and maintain frequency within predefined/acceptable bounds. We evaluate our method on standard transmission systems to empirically validate its ability to localize dominant sources of vulnerabilities. For a 1354-bus large system, our method detects compensations to only four buses under N-1 generation outage (3424.8 MW) while enforcing a maximum allowable steady-state frequency drop of 0.06 Hz (otherwise, frequency drops by nearly 0.08 Hz). We further validate the scalability of our method, requiring less than four minutes to obtain sparse solutions for the 1354-bus system.

Figures (7)

• Figure 1: Primary frequency response
• Figure 2: A few localized compensations can stabilize the entire system. Top row (baseline): simulation of steady-state frequency deviation after generation outage, with droop response considered. Bottom row (proposed method): Red nodes denote the magnitude of compensation sources (in per-unit current) needed at a few identified nodes to restore the $\Delta f$ within bound, localizing and quantifying instability vulnerabilities.
• Figure 3: Droop characteristics of a generator with quadratic relaxation.
• Figure 4: Case30 frequency curve, N-1 generator contingency.
• Figure 5: Case300 frequency curve under N-1 generator contingencies. Cases with $>$ 1500MW loss are not shown in (a) because the system completely collapses and the frequency deviation becomes non-informative. In (b), these cases are recovered to a user-defined healthy state through our proposed method.