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Neste Jacobs Oy / Hans Aalto1 ”Transfer Functions for Natural Gas Pipeline Systems” Dr. Hans Aalto

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Neste Jacobs Oy / Hans Aalto2

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3 Main pipeline system components: Compressor stations, pipeline segments and offtakes 50-100 km Compressor station discharge pressures are usually used to operate the pipeline

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Neste Jacobs Oy / Hans Aalto4 The dynamics, i.e. response to discharge pressures (and offtake gas flow changes) is slow or very slow How would we obtain the transfer function between 2 variables of a given pipeline? - Identify from true pipeline system data - Identify from dynamic simulator data “Direct method”: from design data to transfer functions!

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Neste Jacobs Oy / Hans Aalto5 Start with the PDE for a (=each!) pipeline segment This is the simplest isothermal PDE model for pipelines in the horizontal plane only and with small gas velocities!

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Neste Jacobs Oy / Hans Aalto6 Discretize w.r.t to the space co-ordinate z, using N elements (nodes) for each segment (!) i=1,2,….N Compressor station between node “k-1” and “k” : PI- controller of discharge pressure manipulating gas flow: PI

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Neste Jacobs Oy / Hans Aalto7 … Linearize this large ODE model in a given steady state operating point where x ^ [ΔP 1 Δq 1 ΔP 2 Δq 2 … ΔP N Δq N ] T or:

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Neste Jacobs Oy / Hans Aalto8 Matrices A (2Nx2N) and B (2Nxm) depend on the geometry, physical parameters, node partition and steady state data = design (engineering) information C (1x2N) is needed just to select which state variable is of interest The rest is easy, obtain the transfer function from (A,B,C) using standard methods ??? eg. ss2tf of Matlab

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Neste Jacobs Oy / Hans Aalto9 NO! Transfer function from large system is difficult, even if dominating time constants may be obtained. In our case, numerator dynamics has relevance!

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Neste Jacobs Oy / Hans Aalto10 => Use Linear Model Reduction techniques! Truncation: Solve P and Q from Compute Hankel singular values Arrange eigenvectors of PQ into: The upper N r <<2N submatrices of: provide a greately reduced linear state space system

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Neste Jacobs Oy / Hans Aalto11 Balanced truncation: P and Q required to be diagonal... Transfer function from reduced model with N r = 3…4 is easily obtained with standard methods!

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Neste Jacobs Oy / Hans Aalto12 Pipeline system w. 6 segments, 8 offtakes, 4 compressor stations and 70 nodes => 2N=140 CS1 CS2 CS4 10 20 2 3 4 5 6 7 10 20 30 15 45 Pb Pa

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Neste Jacobs Oy / Hans Aalto13 Transfer function from CS2 discharge pressure to “Pa”, far downstream CS2 [time constant]=minutes!: ~ Dito for “Pb”, close to CS2:

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Neste Jacobs Oy / Hans Aalto14 Compare nonlinear ODE model and linear reduced order model (step response of Pa to CS2) Time, minutes Pa [bar] deviation from steady state

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Neste Jacobs Oy / Hans Aalto15 Further development - Non-isothermal pipeline - High gas speed - Vertical direction - Reduce nonlinear ODE model first, then linearize = Proper Orthogonal Decomposition LOPPU!

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