Random load fluctuations and collapse probability of a power system operating near codimension 1 saddle-node bifurcation

For a power system operating in the vicinity of the power transfer limit of its transmission system, effect of stochastic fluctuations of power loads can become critical as a sufficiently strong such fluctuation may activate voltage instability and lead to a large scale collapse of the system. Considering the effect of these stochastic fluctuations near a codimension 1 saddle-node bifurcation, we explicitly calculate the autocorrelation function of the state vector and show how its behavior explains the phenomenon of critical slowing-down often observed for power systems on the threshold of blackout. We also estimate the collapse probability/mean clearing time for the power system and construct a new indicator function signaling the proximity to a large scale collapse. The new indicator function is easy to estimate in real time using PMU data feeds as well as SCADA information about fluctuations of power load on the nodes of the power grid. We discuss control strategies leading to the minimization of the collapse probability.

💡 Research Summary

The paper investigates how random fluctuations of electrical loads can trigger voltage instability and large‑scale blackouts when a power system operates close to the power‑transfer limit of its transmission network, i.e., near a codimension‑1 saddle‑node bifurcation. The authors begin by linearizing the full nonlinear swing‑equation model around an operating point and identifying the eigenmode whose eigenvalue λ₁ approaches zero at the bifurcation. This “slow mode” dominates the dynamics because all other eigenvalues remain well‑damped.

Random load variations are modeled as Gaussian white noise ξ(t) with intensity σ, which is realistic for short‑term fluctuations captured by SCADA or PMU data. Projecting the noise onto the slow mode yields a one‑dimensional stochastic differential equation:

  ẏ(t) = λ₁ y(t) + σ_v ξ(t)

where y(t) is the coordinate of the slow mode and σ_v quantifies how strongly the load noise excites this mode. Solving this linear SDE, the authors derive an explicit autocorrelation function

  C(τ) = (σ_v² / 2|λ₁|) e^{‑|λ₁|τ}

which shows that the correlation time τ_c = 1/|λ₁| diverges as λ₁ → 0. This mathematical result explains the well‑known phenomenon of critical slowing‑down: near the voltage‑stability limit, voltage fluctuations become slower and larger in amplitude, a pattern that can be observed directly in high‑resolution PMU streams.

Next, the paper treats voltage collapse as a first‑passage‑time problem. The system collapses when the slow‑mode coordinate y exceeds a critical threshold y_c that corresponds to the voltage limit V_th. Using the Fokker‑Planck (or equivalently the backward Kolmogorov) equation, the authors obtain an analytical approximation for the mean clearing time (average time to collapse)

  ⟨T⟩ ≈ (2π / √{|λ₁| D}) exp