Equilibrium-Independent Passivity of Power Systems: A Link Between Classical and Two-Axis Synchronous Generator Models
Takayuki Ishizaki, Taku Nishino, Aranya Chakrabortty
TL;DR
This paper develops an energy-based EI-passivity framework for multi-machine power systems using two-axis generator models, framing the network as a feedback interconnection of a linear mechanical subsystem and a nonlinear electromagnetic subsystem. It proves that lossless transmission is necessary for EI passivity, and shows that the convex domain of the strain energy function $U(z)$ defines the largest EI-passive equilibrium set $ $, with convexity tied to the stability of flux-linkage dynamics and to a link with the classical model via SPA through the reduced energy $ ilde U(oldsymbol heta)$. The two-axis and classical generator models become EI-passive under the same energy-convexity condition, with $L_0(oldsymbol heta^{ullet})= abla^2 ilde U(oldsymbol heta^{ullet})$ acting as a generalized synchronizing-torque matrix; the results extend to lossy networks by replacing convexity with positive semidefiniteness of $L_0$. Numerical simulations on the IEEE 9-bus system corroborate that the convex energy domain aligns closely with the set of stable equilibria, validating the practical relevance for stability assessment and control design in heterogeneous power networks.
Abstract
We study the equilibrium-independent (EI) passivity of a nonlinear power system composed of two-axis generator models. The model of our interest consists of a feedback inter-connection of linear and nonlinear subsystems, called mechanical and electromagnetic subsystems. We mathematically prove the following three facts by analyzing the nonlinear electromagnetic subsystem. First, a lossless transmission network is necessary for the EI passivity of the electromagnetic subsystem. Second, the convexity of a strain energy function characterizes the largest set of equilibria over which the electromagnetic subsystem is EI passive. Finally, we prove that the strain energy function for the network of the two-axis generator models is convex if and only if its flux linkage dynamics is stable, and the strain energy function for the network of the classical generator models derived by singular perturbation approximation of the flux linkage dynamics is convex. Numerical simulation of the IEEE 9-bus power system model demonstrates the practical implications of the various mathematical results. In particular, we validate that the convex domain of the strain energy function over which the electromagnetic subsystem is EI passive is almost identical to the set of all stable equilibria. This result is also generalized to lossy power systems based on our finding that the convexity of the strain energy function is equivalent to the positive semidefiniteness of a synchronizing torque coefficient matrix.