This paper explains a graph-based way to model energy systems that span electricity, heat, and mechanics and that operate over very different speeds. The authors present a framework that treats a system as a network of energy storage points linked by power flows. The approach was developed and validated over more than a decade across several universities and companies, and the paper also describes an open-source MATLAB-based toolbox to build and analyze these models.

The authors give a mathematical description of the method and show how it applies to several concrete component types from recent work: single-phase thermal systems, two-phase thermal systems (like refrigeration cycles), and electro-mechanical components. They also survey how the models have been used in the literature. Table I in the paper groups past work by energy domain (electrical, mechanical, thermal) and Table II lists applications such as architecture optimization, control co-design, design optimization, estimation and fault detection, and model-based control.

At a high level the models use a graph made of vertices and edges. Vertices represent places where energy is stored and can be either dynamic states or external inputs. Edges represent instantaneous power flow between vertices. The network is written with an incidence matrix that encodes which vertices connect to which edges. In simple form the dynamic equations look like C ẋ = −M P, where C is a matrix of storage capacities, x are the dynamic states, M is the incidence matrix, and P is the vector of power flows. Each power flow is a function of the states at the two ends of its edge and any control inputs.

This graph view matters because it keeps the physical conservation of energy explicit while also revealing the system’s structure. That structure makes it easier to do symbolic analysis, to split large models into smaller parts using graph tools, and to swap components in and out — a “plug-and-play” style useful for design and co-design of controllers. The authors emphasize that the method helps with stiff dynamics that arise when parts of a system evolve on very different timescales (for example, voltages changing in sub-milliseconds while large thermal storage changes over hours). They also report successful uses in decentralized and hierarchical Model Predictive Control (MPC), among other control approaches.