A Unified (P)DAE Modeling Approach for Flow Networks

Jansen, Lennart; Tischendorf, Caren

doi:10.1007/978-3-662-44926-4_7

Cited by 16 publications

(14 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this generality, flow networks arise in many different application fields: let us mention for example distribution networks [11,14,27,43], supply chains [17], traffic networks [10,21], power systems [9,18], inventory systems [4,8], etc. We do not at all attempt to be exhaustive, and references are only a small sample of a huge amount of literature.…”

Section: Flowsmentioning

confidence: 99%

“…From different perspectives, this kind of problems have been addressed in many application fields, which include coupled oscillators, operations research, circuit theory, electronic engineering, water and gas networks, power systems, traffic networks or multiagent systems, to name but a few: cf. [2,3,9,10,11,16,21,26,27,28,32,34,37,42,48]. Our point of view is deliberately a general one, as we aim to explore in structural terms certain properties of dynamical systems defined on networks, without focusing on specific models coming from applications; in our analysis we examine systematically the way in which the dynamical processes interact with the graph-theoretic underlying structure, looking for general results except for the fact that the dynamics is assumed to be defined by a potential-driven flow.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Structure and stability of the equilibrium set in potential-driven flow networks

Riaza

2017

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

In this paper we address local bifurcation properties of a family of networked dynamical systems, specifically those defined by a potential-driven flow on a (directed) graph. These network flows include linear consensus dynamics or Kuramoto models of coupled nonlinear oscillators as particular cases. As it is well-known for consensus systems, these networks exhibit a somehow unconventional dynamical feature, namely, the existence of a line of equilibria, following from a well-known property of the graph Laplacian matrix in connected networks with positive weights. Negative weights, which arise in different contexts (e.g. in consensus models in signed graphs or in Kuramoto models with antagonistic actors), may on the one hand lead to higher-dimensional manifolds of equilibria and, on the other, be responsible for bifurcation phenomena.In this direction, we prove a saddle-node bifurcation theorem for a broad family of potential-driven flows, in networks with one or more negative weights. The goal is to state the conditions in structural terms, that is, in terms of the expressions defining the flowrates and the graph-theoretic properties of the network. Not only the eigenvalue requirements but also the nonlinear transversality assumptions supporting the bifurcation motivate an analysis of independent interest concerning the rank degeneracies of nodal matrices arising in the linearized dynamics; this analysis is performed in terms of the contraction-deletion structure of spanning trees and uses several results from matrix analysis. Different examples illustrate the results; some linear problems (including signed graphs) are aimed at illustrating the analysis of nodal matrices, whereas in a nonlinear framework we apply the characterization of saddle-node bifurcations to networks with a sinusoidal (Kuramoto-like) flow.

show abstract

Section: Flowsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Structure and stability of the equilibrium set in potential-driven flow networks

Riaza

2017

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

show abstract

“…Section 20: Dynamics and control the overall transport network equation system is given by [4] A P l q P l + A P r q P r + A…”

Section: Water Transport Network Modelingmentioning

confidence: 99%

A Topology Based Discretization of PDAEs Describing Water Transportation Networks

Huck

Jansen

Tischendorf

2014

Proc Appl Math and Mech

Self Cite

View full text Add to dashboard Cite

We consider partial differential algebraic systems (PDAEs) describing water transportation networks. Similar to the approach in [6], we follow the method of lines for the discretization. However, we do not consider free surface flow models but pressure flow models covering hydraulic shocks. Moreover, we include switching models reflecting the on/off state of pumpes and valves. Aiming at a stable numerical simulation of the PDAEs we present a topology based spatial discretization that results in a differential algebraic system (DAE) of index 1. Furthermore we show that the DAE index can be higher than 1 if the spatial discretization is not adapted to the position of reservoirs and demand nodes within the network.Let the water network graph have n E edges and n N nodes. Then, the relation between the edges and the nodes of the network graph can be easily described by the incidence matricesThe well-known incidence matrix A for network graphs is given by A = A l + A r . The flow balance at each node provideswith q E l and q E r describing the flows at the left and right ends of the edges. Here, we use the convention that each edge directs from its left to its right end. The flow vector q N contains the in/out flows at the nodes. We consider water transport networks consisting of junctions (J), tanks (T) and reservoirs (S) as nodal model elements as well as pipes (P) and valves/pumps (R) as edge model elements satisfying the characteristic equationsHere, q set is the given demand at the junctions (equals zero at each node without a demand), C is the tank capacity, p set is the given pressure at the reservoirs, f R is a monotone, given function describing the resistance of valves/pumps. The switching parameter s is time-dependent. It equals 1 if the valve/pump is open/on and zero otherwise. In transport pipes one has usually only velocities up to 5m/s. Therefore, we can neglect the convective terms of the Euler equations and consider the pipe model (see [2])with the pipe diameter d, the pipe cross-section area a, the pipe wall sound velocity c, the fluid density , the Darcy friction factor λ, the gravitation acceleration g and the pipe slope angle α. Equation (1.2) describes the continuity equation and the mass flow balance through the pipe. We do not assume ∂ x q P to be constant in order to cover also hydraulic shocks. The boundary conditions are given as p(x l ) = p P l , p(x r ) = p P r , q(x l ) = q P l , q(x r ) = q P r .(1.3)Combining the equations (1.1)-(1.3) and using the splitting

show abstract

“…Besides energy hubs, there is a number of simulation-oriented approaches for modeling multi-carrier energy systems. In their most general form these approaches lead to differential-algebraic equations (DAEs) or partial DAEs (Jansen and Tischendorf, 2014) that are not applicable for control design. Martinez-Mares and Fuerte-Esquivel (2012) and Sirvent et al (2017) propose simplified steady state models for coupled electricity and gas transmission systems.…”

Section: Introductionmentioning

confidence: 99%

Towards Port-Hamiltonian Modeling of Multi-Carrier Energy Systems: A Case Study for a Coupled Electricity and Gas Distribution System

et al. 2018

View full text Add to dashboard Cite

Abstract:Multi-carrier energy systems have been identified as a major concept for future energy supply. For their operation, model-based control methods are necessary whose design requires modular, multi-physical control-oriented models. In literature, there exists no control design model which combines the variables of the networks and system dynamics that go beyond ideal storage elements. Port-Hamiltonian systems represent a promising approach for the scalable modeling and control of multi-carrier energy systems. In this publication we present a case study which illustrates the port-Hamiltonian modeling of an exemplary coupled electricity and gas distribution system. Simulations indicate the plausibility of the presented model.

show abstract

A Unified (P)DAE Modeling Approach for Flow Networks

Cited by 16 publications

References 27 publications

Structure and stability of the equilibrium set in potential-driven flow networks

Structure and stability of the equilibrium set in potential-driven flow networks

A Topology Based Discretization of PDAEs Describing Water Transportation Networks

Towards Port-Hamiltonian Modeling of Multi-Carrier Energy Systems: A Case Study for a Coupled Electricity and Gas Distribution System

Contact Info

Product

Resources

About