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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

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In the 4 th page, the incorrect formulae S := 1 2 Dπ 2 which was inadvertently published is now corrected as S := 1 4 πD 2 .

One challenge for the simulation and optimization of real gas pipe networks is the treatment of compressors. Their behavior is usually described by characteristic diagrams reflecting the connection of the volumetric flow and the enthalpy change or shaft torque. Such models are commonly used for an optimal control of compressors and compressor stations [4,7] using stationary models for the gas flow through the pipes. For transient simulations of gas networks, simplified compressor models have been studied in [1][2][3]. Here, we present a transient simulation of gas pipe networks with characteristic diagram models of compressors using a stable network formulation as (partial) differential-algebraic system. Let G = (V, E) a directed graph with vertices V = V + ∪ V − and edges E = E P ∪ E C where V − and V + are the nodes where gas can enter and exit the network respectively and E P and E C being the set of pipes and compressors respectively. In addition we define the sets δ − (u) and δ + (u) as the set of edges that are directed towards and away from node u ∈ V respectively.Theorem 1.1 Let G = (V, E) be a connected, directed graph that describes a gas network with pipes and compressors and letThen it holds, that the pipes can be directed in a way that Pipe ModelingWe model pipes by a simplification of the isothermal Euler equations [2] ∂ t p e + c 2 a e ∂ x q e = 0, ∂ t q e + a e ∂ x p e = − λ e c 2 2D e a e q e |q e | p e − ga e h e c 2 p e , e ∈ E P (1) on [0, T ] × [0, e ] where p and q are the pressure and mass flow along pipe e, a e is the cross-sectional area, D e the diameter, λ e the friction factor, g the gravitational acceleration, h e the elevation of the pipe, c is the speed of sound, e the length of the pipe. Also we identify the point x = 0 with the position at the node u and x = e with node v for e = (u, v). Such a modeling is known to describe the gas flow through a pipe sufficiently well if the velocity of the gas is much less than the speed of sound which is usually the case for real gas transport networks. Compressor ModelingFor the compressor model we consider the characteristic diagram model for turbo-compressors describing the relation between the adiabatic enthalpy and the volumetric flow rate [4,7].Here H e is the adiabatic enthalpy, Q the volumetric flow rate, q e the massflow and n e the speed. Ψ e (Q, n) =Q T e A ene with A e ∈ R 3×3Q e = (1, Q e , Q 2 e ) andn e = (1, n e , n 2 e ) . The fourth equation of (2) determines the control of the compressor. We model the entry-nodes as a boundary condition for the pressure, leading to p e (t, 0) = p Γ u (t) for e ∈ δ + (u), u ∈ V + . For the nodes u ∈ V − we model the coupling by balance equations for the massflows e P ∈δ − (u) q e P (t, e P ) − e P ∈δ + (u) q e P (t, 0) + e C ∈δ − (u) q e C (t) − e C ∈δ + (u) q e C (t) = q Γ u (t)