”Transfer Functions for Natural Gas Pipeline Systems” Dr. Hans Aalto

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

2. Neste Jacobs Oy / Hans Aalto

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 Neste Jacobs Oy / Hans Aalto

4. 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! Neste Jacobs Oy / Hans Aalto

5. 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! Neste Jacobs Oy / Hans Aalto

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

7. … Linearize this large ODE model in a given steady state operating point or: where x^ [ΔP1Δq1ΔP2Δq2 … ΔPNΔqN ]T Neste Jacobs Oy / Hans Aalto

8. Matrices A (2Nx2N) and B (2Nxm) depend on the geometry, physical parameters, node partition and steady state data = design (engineering) informationC (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 Neste Jacobs Oy / Hans Aalto

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

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

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

12. 30 20 5 6 CS4 CS1 20 10 CS2 45 15 3 2 4 Pb 10 7 Pa 20 Pipeline system w. 6 segments, 8 offtakes, 4 compressor stations and 70 nodes=> 2N=140 Neste Jacobs Oy / Hans Aalto

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

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

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