A Stiffness-Oriented Model Order Reduction Method for Low-Inertia Power Systems

TL;DR

The paper addresses the stiffness challenge in low-inertia power systems dominated by inverter-based resources, which complicates time-domain simulation and control design. It introduces a stiffness-oriented model order reduction method that rotates the linearized system into a decoupled basis using a real Jordan form and truncates fast modes, with the same transformation applied to the nonlinear system to obtain an approximate slow-fast separation. The method preserves the slow eigenvalues of the linearized system, reduces stiffness, and enables accurate integration with substantially larger time steps, demonstrated on a 3-bus test with mixed converter and synchronous-machine units and extended to larger networks. The results show speed-ups up to about 100x compared with the full model, offering a scalable tool for simulation-driven control and estimation in future low-inertia grids.

Abstract

This paper presents a novel model order reduction technique tailored for power systems with a large share of inverter-based energy resources. Such systems exhibit an increased level of dynamic stiffness compared to traditional power systems, posing challenges for time-domain simulations and control design. Our approach involves rotation of the coordinate system of a linearized system using a transformation matrix derived from the real Jordan canonical form, leading to mode decoupling. The fast modes are then truncated in the rotated coordinate system to obtain a lower-order model with reduced stiffness. Applying the same transformation to the original nonlinear system results in an approximate separation of slow and fast states, which can be truncated to reduce the stiffness. The resulting reduced-order model demonstrates an accurate time-domain performance, the slow eigenvalues of the linearized system are correctly preserved, and a reduction in the model stiffness is achieved, allowing for accurate integration with increased step size. Our methodology is assessed in detail for a 3-bus system with generation units involving grid-forming/following converters and synchronous machines, where it allows for a computational speed-up of up to 100x compared to the original system. Several standard larger test systems are also considered.

Figures (6)

• Figure 1: General converter configuration scheme.
• Figure 2: Single line diagram of the considered 3-bus system.
• Figure 3: Eigenvalue spectrum of the linearized full and reduced models for Scenario 1 (left column), Scenario 2 (middle column), and Scenario 3 (right column). The upper row shows the eigenvalue spectrum of the reduced models obtained using the proposed stiffness-oriented MOR (Section \ref{['sec:proposed_MOR']}) and PFA (Section \ref{['sec:pfa']}) while the lower row shows the one of the models obtained using balanced truncation (Section \ref{['sec:bt']}).
• Figure 4: Time domain solution of distinct slow and fast states for Scenario 1.
• Figure 5: RMSE vs. inverse of step size ($h^{-1}$) for different combinations of models (the reduced model obtained using the proposed stiffness-oriented MOR ($\mathrm{sor}$), the reduced model using PFA ($\mathrm{pfa}$) and the original full model ($\mathrm{full}$) for Scenario 1 (left), Scenario 2 (middle), and Scenario 3 (right)).