In this talk we discuss discrete port-Hamiltonian systems from the view point of geometric numerical integration. Our starting point is a continuous port-Hamiltonian system which is discretized using a passivity-preserving numerical time integration method. We propose two different numerical approaches. We analyse these methods, focusing in particular on discrete energy-preserving and passivity-preserving interconnection of simpler systems. Connections to nonholonomic systems and their energy-preserving integration are considered. For nonholonomic systems, a (numerical) comparison with Lagrange-d’Alembert variational integrators will be presented.

Energy based modeling, simulation and control using  port-Hamiltonian descriptor systems

Riccardo Morandin, Inst. f. Mathematik, TU Berlin

Recent developments in the energy market require new mathematical models and suitable algorithms for the efficient usage of the existing networks. Starting from these real-world problems, it is necessary to concentrate on methods for large-scale energy networks and, in particular, address optimization and stability analysis, model predictive control, model-order reduction, uncertainty quantification, and related topics on energy networks. The abstract setting allows for consideration of applications arising from both, gas and power networks.

Our framework of choice is the one of port-Hamiltonian descriptor systems, or pHDAEs. These are energy-based equations with a special structure, that guarantees several beneficial properties, e.g. inherent stability and passivity, physical interpretation of variables, simple interconnection, structure-preserving model reduction, robust numerical integration and simplification of feedback stabilization.

In this talk, we present a new formulation for port-Hamiltonian descriptor systems, which extends previous models by including under- and overdetermined systems, and arbitrary differentiable Hamiltonian functions. Every system of this form is associated with a geometric object known as Dirac structure, that encloses its energy balance properties. Furthermore, we investigate the application of collocation methods for time-discretization to these systems and we show that, under certain assumptions, structure preservation can be achieved. Finally, we show one example from the power network setting and some numerical results.