Horizons of mathematics, with Volker Mehrmann (TU Berlin, Germany)
Port-Hamiltonian systems make it possible to describe efficiently how energy flows and is exchanged between different parts of a system and with the environment. Volker Mehrmann (Technical University of Berlin, Germany), a specialist in these systems, has applied them to model district heating and gas transport networks. “Using this new modelling framework, we were able to run simulations much faster and solve optimisation problems that had previously been unsolvable,” he says. On 16 January 2026, at 13:00, he will present his results at the next mathematics colloquium jointly organised by ICMAT, the Autonomous University of Madrid, the Complutense University of Madrid and Carlos III University of Madrid. The event will take place at the UC3M Leganés Campus and will also be streamed live.
Volker Mehrmann (TU Berlin) will give the next joint ICMAT-UAM-UC3M-UCM colloquium. Image courtesy of Mehrmann.
Laura M. Iraola (ICMAT-CSIC)
One of the classical tools of mathematical physics is the Hamiltonian, which describes how the energy of a system evolves without losses. However, many physical systems are open and interconnected. That is, they store energy but also dissipate it and exchange it with the environment. Port-Hamiltonian systems are designed precisely to describe these situations.
“Their key property is the energy balance: the energy stored in a physical system is equal to the energy supplied to the system minus the energy dissipated to the environment,” explains Volker Mehrmann, Professor at the Technical University of Berlin (TU Berlin), Germany.
For example, to describe and optimise a district heating network, this approach represents the thermal energy of the water as the key quantity of the system, models the pipes as elements that transport heat with losses, and the buildings and the generation plant as ports through which energy enters and leaves. In this way, the model always maintains a consistent energy balance and makes it easier to analyse, control and optimise the operation of the entire network in a clear manner.
Mehrmann and his team have derived these models of the district heating network in the city of Ludwigshafen in cooperation with the Fraunhofer ITWM in Kaiserslautern and colleagues from the University of Trier. “The aim was to model and optimise the flow of hot water in the network,” explains Mehrmann. “Using the new modelling framework, we were able to have much faster simulation times and we could solve optimization problems that were so far not solvable”, he adds. They have also applied this approach to gas transport networks.
The key advantage is that, with this approach, automatized modelling is possible (following clear rules based on the energy of the system) and a large system can be divided into smaller subsystems, which are modelled separately and then interconnected as a network. “This makes mathematical modelling extremely convenient and the models very robust to uncertainties”, says Mehrmann.
This new approach opens up a wide range of possible applications: circuit simulation, power grids, robotics or even biomedical applications, such as the interaction of the brain with the surrounding fluid.
On 16 January, Mehrmann will describe this framework and analyse its properties from an algebraic, analytical and geometrical point of view at the forthcoming colloquium “Mathematical modelling, analysis and numerical methods for modern energy transport networks”, jointly organised by the Institute of Mathematical Sciences (ICMAT), the Autonomous University of Madrid (UAM), the Complutense University of Madrid (UCM) and Carlos III University of Madrid (UC3M). It will take place at 13:00 in the Salón de Grados of the Padre Soler Building, at the UC3M Leganés Campus, and will also be streamed live.
Mathematical models with algebraic and geometrical–differential properties
Port-Hamiltonian systems were introduced around 30 years ago in the field of systems and control theory. “The formulation is based originally on the differential-geometric concepts of Dirac and Lagrangian structures, that allow to describe systems as differential equations on manifolds respectively locally via linear spaces that are associated with symmetry and skew symmetry of matrices, matrix functions and operators”, explains Mehrmann. All of this provides structural properties such as stability or passivity of the systems
Mehrmann came to this field of research thanks to the award in 2011 of an Advanced Grant from the European Research Council (ERC) to work on modelling, simulation and control of multiphysics systems. “Adopting this modelling framework was the key to the success of the project”, he admits
“By encoding the physics in the mathematical model, we gain a better understanding of the physics through the mathematical properties,” he says. “At the same time, we have better simulation and optimization methods”, he concludes
Volker Mehrmann
Volker Mehrmann (Detmold, Germany, 1955) is a specialist in the fields of numerical linear algebra, differential equations and control theory. He obtained his PhD from Universität Bielefeld (Germany) in 1982 and has developed his research career at centres and universities such as Kent State University (USA), the University of Wisconsin (USA), the IBM Research Centre in Heidelberg (Germany), RWTH Aachen University (Germany) and Chemnitz University of Technology (TU Chemnitz, Germany). In 2000, he joined the Institute of Mathematics at the Technical University of Berlin (Technische Universität Berlin), where he has worked ever since
In 2011, he received an Advanced Grant from the European Research Council (ERC) for the modelling, simulation and control of multiphysics systems
In addition, Mehrmann has received various distinctions for his scientific contributions, including his appointment in 2022 as a Fellow of the American Mathematical Society (AMS). In 2015, he received the Gauss Lectureship from the German Mathematical Society, in 2018 he was awarded the W T and Idalia Reid Prize from the Society for Industrial and Applied Mathematics (SIAM), and in 2019 he was recognised with the Hans Schneider Prize in Linear Algebra from the International Linear Algebra Society
He has also served as President of mathematical associations such as GAMM (International Association of Applied Mathematics and Mechanics) from 2011 to 2013, and of the European Mathematical Society (EMS) from 2019 to 2022. He is a member of the German Academy of Science and Engineering (Acatech), the European Academy of Sciences and of Academia Europaea.
ICMAT-UAM-UC3M-UC3M Joint Colloquium: “Mathematical modelling, analysis and numerical methods for modern energy transport networks”, Volker Mehrmann (TU Berlin)”
Date & time: Friday, 16 January 2026, 13:00
Venue: Salón de Grados del Edificio Padre Soler, Campus de Leganés, Universidad Carlos III de Madrid.
Abstract: Mathematical models in form of differential equations are used in all areas of science and engineering for the simulation, control and optimization of real-world physical systems. Ideally such models are close to the real physical system in the sense that the physical properties are encoded in the mathematical structure of the equations. The model equations should be easy to analyse and allow for good numerical methods that reflect the physical properties as well. A modern modelling paradigm is that of port-Hamiltonian systems, which originates from Hamilton dynamics enriched with inputs, outputs and constraints. We will describe this framework and analyse its properties from an algebraic, analytic and geometric point of view. We will illustrate the successful use of the model class at the hand of district heating and gas transport networks modelled by the one-dimensional Euler equations.