Nonlinear Model Order Reduction of Power Grid Networks using Quadratic Manifolds

TL;DR

The paper addresses the computational burden of transient stability analysis in large-scale power grids by introducing a nonlinear projection-based MOR that uses a quadratic manifold to augment a POD basis with a learned quadratic correction. A regularized least-squares procedure constructs the quadratic mapping, enabling accurate modeling of nonlinear rotor-swing dynamics without increasing reduced coordinates. The approach is validated on benchmark systems (IEEE 118-bus, IEEE 300-bus, and Polish 2736-bus) under normal and fault conditions, showing improved trajectory accuracy and higher snapshot-energy retention compared with linear POD, albeit with some online-cost overhead. The work provides a practical algorithm and publicly available code, offering a compact, data-driven surrogate suitable for fast dynamic simulations and online stability monitoring.

Abstract

The increasing size and complexity of modern power systems have led to a high-dimensional mathematical model for transient stability studies, rendering full-scale simulations computationally burdensome. While dimensionality reduction is essential for reducing this complexity, conventional approaches in power systems predominantly rely on linear projection methods. Such linear subspaces have limited capability for representing the inherently nonlinear swing dynamics of synchronous machines, often resulting in poor approximations and inefficient compression. To address these limitations, this paper introduces a quadratic manifold-based model order reduction (MOR) framework to accelerate the transient dynamic simulations in power systems. The proposed method combines the linear proper orthogonal decomposition (POD) basis with a learned quadratic correction term that minimizes the reconstruction error. This yields a scalable MOR strategy capable of handling strongly nonlinear behaviors, particularly those arising during fast-acting faults, where linear techniques typically fail. The method is tested on a range of benchmark power system models of increasing size and complexity. In addition, we provide a detailed numerical algorithm for constructing the quadratic manifold, along with the corresponding implementation code.

Figures (6)

• Figure 1: Graphical illustrations of linear and nonlinear projection-based model order reduction techniques. The solid red line represents the true solution trajectory, and the dashed black line represents the approximated solution captured on the manifolds.
• Figure 2: Simulation results of IEEE 118 bus system: (A) variation of $\delta_{avg}$ with respect to time; (B) variation of $\Delta\omega_{avg}$ with respect to time; (C) snapshot energy captured by the system at various reduced basis dimensions; (D) comparison of relative error for linear and quadratic manifold ROMs; (E) comparison of the CPU times for increasing basis dimension; (F) comparison of the output response for three individual generators with respect to time; (G) comparison os system trajectory onto a linear manifold and quadratic manifold; (H) evolution of angular velocities with respect to time
• Figure 3: Simulation results of IEEE 300 bus system: (A) variation of $\delta_{avg}$ with respect to time; (B) variation of $\Delta\omega_{avg}$ with respect to time; (C) snapshot energy captured by the system at various reduced basis dimensions; (D) comparison of relative error for linear and quadratic manifold ROMs; (E) comparison of the CPU times for increasing basis dimension; (F) comparison of the output response for three individual generators with respect to time; (G) comparison os system trajectory onto a linear manifold and quadratic manifold; (H) evolution of angular velocities with respect to time
• Figure 4: Simulation results of Polish 2736-bus system: (A) variation of $\delta_{avg}$ with respect to time; (B) variation of $\Delta\omega_{avg}$ with respect to time; (C) snapshot energy captured by the system at various reduced basis dimensions; (D) comparison of relative error for linear and quadratic manifold ROMs; (E) comparison of the CPU times for increasing basis dimension; (F) comparison of the output response for three individual generators with respect to time; (G) comparison os system trajectory onto a linear manifold and quadratic manifold; (H) evolution of angular velocities with respect to time
• Figure 5: Simulation results for fault case 1: (Left) average deviation of $\Delta\omega(t)$; (Right) comparison of relative errors for linear and quadratic manifolds