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Qualitative stability and synchronicity analysis of power network models in port-Hamiltonian form

Mehrmann, Volker; Morandin, Riccardo; Olmi, Simona; Schöll, Eckehard

FG Nichtlineare Dynamik und Kontrolle

In view of highly decentralized and diversified power generation concepts, in particular with renewable energies, the analysis and control of the stability and the synchronization of power networks is an important topic that requires different levels of modeling detail for different tasks. A frequently used qualitative approach relies on simplified nonlinear network models like the Kuramoto model with inertia. The usual formulation in the form of a system of coupled ordinary differential equations is not always adequate. We present a new energy-based formulation of the Kuramoto model with inertia as a polynomial port-Hamiltonian system of differential-algebraic equations, with a quadratic Hamiltonian function including a generalized order parameter. This leads to a robust representation of the system with respect to disturbances: it encodes the underlying physics, such as the dissipation inequality or the deviation from synchronicity, directly in the structure of the equations, and it explicitly displays all possible constraints and allows for robust simulation methods. The model is immersed into a system of model hierarchies that will be helpful for applying adaptive simulations in future works. We illustrate the advantages of the modified modeling approach with analytics and numerical results. To reach the goal of temperature reduction to limit the climate change, as stipulated at the Paris Conference in 2015, it is necessary to integrate renewable energy sources into the existing power networks. Wind and solar power are the most promising ones, but the integration into the electric power grid remains an enormous challenge due to their variability that requires storage facilities, back-up plants, and accurate control processing. The current approach to describe the dynamics of power grids in terms of simplified nonlinear models, like the Kuramoto model with inertia, may not be appropriate when different control and optimization tasks are needed to be addressed. Under this aspect, we present a new energy-based formulation of the Kuramoto model with inertia that allows for an easy extension if further effects have to be included and higher fidelity is required for qualitative analysis. We illustrate the new modeling approach with analytic results and numerical simulations carried out for a semi-realistic model of the Italian grid and indicate how this approach can be generalized to models of finer granularity.
  • This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos 28, 101102 (2018) and may be found at https://doi.org/10.1063/1.5054850.