We present a unified topological analysis for physical networks coupled to black-box models. While the governing equations of the physical network are derived by representing the network as a linear graph, the equations of the black-box model are characterized as an abstract class of Differential-Algebraic equations (DAEs) of a specific form (Functional Mock-up Unit). Combining those two representations, existence and uniqueness results can be formulated based on the connectivity of the physical network and a concrete classification of the black-box models. This is the natural extension of the approach derived in [3].
Black-box models in system simulationThis work is related with a commercial software allowing to develop and test automotive components virtually on the desktop or in a testbed with hardware components. Its modular drag-and-drop approach makes modeling very simple, however, creates large and entangled mathematical models. To aid the customer in the modeling and provide a suitable model for a fast and accurate simulation, a topology based approach is pursued. As a case study, cooling and lubrication circuits, which are modeled as liquid flow networks, are considered. Combining graph and DAE-theory, structural solvability conditions can be derived, which allow a plausibility check and constructive error handling prior to the simulation. Furthermore a surrogate model, that is free of redundant and hidden information, can be constructed. This approach ensures an accurate and efficient numerical solution, cp. [2,3] .A natural extension of the latter mentioned approach is the incorporation of black-box models, i.e., physical or software components with unknown internal structures and principles. Black-box models allow to incorporate external or customerdefined components into the software while guarding the intellectual property, cp. [1]. Within this work, the focus is on black-boxes that comply with the FMI standard, cp. [4]; i.e. we consider Functional Mock-up Units (FMUs). Mathematically, FMUs can be repesented as the following set of equations. Find x and y, such thatfor given time t, input u and initial condition x 0 for the state x. Here f and g are not known explicitly, only their values for given t, x and u can be evaluated. The FMUs are coupled to the liquid flow network via defined boundary conditions on the inputs u and the outputs y.
Model of liquid flow networks coupled to FMUsWe consider a liquid flow network N = {PI, PU, J C, DE, RE} filled with an incompressible fluid and composed of pipes PI, pumps PU, junctions J C as well as mass flow and pressure boundary conditions DE and RE. The pipes and pumps carry directed mass flows q Pi , q Pu , while the junctions are equipped with the pressure p Jc . The boundary conditions consist of classical demands DE and reservoirs RE imposing a given mass flow q De and pressure p Re as well as of demand FMUs B DE and reservoir FMUs connected to pipes and pumps, i.e., B REPi and B REPu , imposing a mass flow q BDe and pressures p BRe Pi , p BRe Pu by thei...
