”transfer functions for natural gas pipeline systems” dr. hans aalto

Neste Jacobs Oy / Hans Aalto1

”Transfer Functions for Natural Gas Pipeline Systems”

Dr. Hans Aalto


Main pipeline system components:Compressor stations, pipeline segments and offtakes

50-100 km

Compressor station discharge pressures are usually used to operate the pipeline


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!


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!

0P b q

t A z

2

0q P b q

A ft z DA P


Discretize w.r.t to the space co-ordinate z, using N elements (nodes) for each segment (!) i=1,2,….N

1( )i ii i

i i

dP bq q

dt A z

2

1( )i i i ii i i

i i i i

dq A b qP P f

dt z D A P

Compressor station between node “k-1” and “k” : PI-controller of discharge pressure manipulating gas flow:

11 ,( ) ( )k

k k k k SET ki

dq KK q q P P

dt T

PI


… Linearize this large ODE model in a given steady state operating point

where x ^ [ΔP1 Δq1 ΔP2 Δq2 … ΔPN ΔqN ]T

2,

1 2, ,

( ) 2 i SS ii ii i i i i

i SS i SS

q Pd q qP P

dt P P

1( )ii i i

d Pq q

dt

or:( )

( ) ( ),

( ) ( )

d tt t

dtt t

xAx Bu

y Cx


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

( )( ) ( ),

( ) ( )

d tt t

dtt t

xAx Bu

y Cx

The rest is easy, obtain the transfer function from (A,B,C) using standard methods ??? eg. ss2tf of Matlab

1 2 3

( 1)( 1)...( )

( 1)( 1)( 1)...a bK T s T s

G sT s T s T s


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


=> Use Linear Model Reduction techniques!Truncation: Solve P and Q from

=

T T

T T

AP + PA +BB 0

A Q+QA+C C 0

Compute Hankel singular values i iσ λ (PQ)

Arrange eigenvectors of PQ into: 2 2N[ 1T v v v

The upper Nr <<2N submatrices of:provide a greately reduced linear state space system

-1

-1

A T AT

B T B

C CT


Balanced truncation: P and Q required to be diagonal . . .

Transfer function from reduced model with Nr = 3…4 is easily obtained with standard methods!


Pipeline system w. 6 segments, 8 offtakes, 4 compressor stations and 70 nodes=> 2N=140

CS1 CS2 CS4

10

20

2 3 4

56

7

20 10

20 30

15 45

PbPa


Transfer function from CS2 discharge pressure to “Pa”, far downstream CS2 [time constant]=minutes!:

1.44

(190.6 1)s ~

1.44(116.9 1)

(117.7 1)(190.6 1)

s

s s

Dito for “Pb”, close to CS2:

1.16(83.5 1)

(11.2 1)(183.6 1)

s

s s


Compare nonlinear ODE model and linear reduced order model (step response of Pa to CS2)

Time, minutes

0 250 500 750 1000 1250 1500-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Pa [bar] deviation from steady state


Further development

- Non-isothermal pipeline- High gas speed- Vertical direction

- Reduce nonlinear ODE model first, then linearize = Proper Orthogonal Decomposition

LOPPU!

”transfer functions for natural gas pipeline systems” dr. hans aalto

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