Small-Signal Stability Analysis of Power Systems by Implicit Multilinear Models

TL;DR

This work tackles small-signal stability analysis in converter-dominated power systems by modeling dynamics with implicit multilinear iMTI models and representing them as CP1-decomposed tensors, enabling efficient linearization to a descriptor form.Exact symbolic multilinearization is used to convert converter nonlinearities (including trigonometric functions) into multilinear representations, with a Lie-Bläcklund transformation establishing equivalence to the original dynamics.Stability is assessed from the generalized eigenvalues of the linear descriptor system obtained from the iMTI model, and the approach is validated on a 3-bus network by comparing time-domain simulations and small-signal results against a nonlinear model, showing close agreement and computational advantages.The methodology supports scalable analysis of modern low-inertia grids and points to future work on larger networks, lower-rank tensor representations, and richer nonlinearities such as saturations, expanding the practical applicability of tensor-based stability analysis in power systems.

Abstract

This paper proposes a new approach to perform small-signal stability analysis based on linearization of implicit multilinear models. Multilinear models describe the system dynamics by multilinear functions of state, input, and algebraic variables. Using suitable transformations of variables, they can also represent trigonometric functions, which often occur in power systems modeling. This allows tensor representations of grid-following and grid-forming power converters. This paper introduces small-signal stability analysis of equilibrium points based on implicit multilinear models using generalized eigenvalues. The generalized eigenvalues are computed from linear descriptor models of the linearized implicit multilinear model. The proposed approach is tested using a 3-bus network example, first by comparing time-domain simulations of the implicit multilinear model with those of the nonlinear model, and second by comparing the generalized eigenvalues with those of the linearized nonlinear model. The results show that the decomposed tensor representation of the implicit multilinear model allows for a faster linearization compared to conventional methods in MATLAB Simulink.

Figures (10)

• Figure 1: Monomial tensor of the Example \ref{['ex:monomial']}
• Figure 2: Phase-locked loop
• Figure 3: Scheme of the essential system in the global frame $\theta_\mathrm{grid}$ (dashed line), and the local frames of the converters (dashed-dotted line)
• Figure 4: Cascaded control loop structure of the grid-forming converter
• Figure 5: Control block diagram of the grid-following converter

Theorems & Definitions (15)

• Definition 2.1
• Example 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Example 2.2
• Example 2.3
• Example 3.1
• Example 3.2: PLL
• Definition 4.1