27876677 gas reduction station model

Modelling of a two-stage high-pressure gas reduction station

I. Fletcher, C. S. Cox, W. J. B. Arden and A. Doonan

Control Systems Centre, SEAT, Urziuersity of Sunderland, Sunder-land, E&and

The delivery of gas from the national transmission system into the regional supergrids, at the correct pressure, is accomplished using a facility referred to in the industry as an above-ground installation (AGI). Early attempts to control such systems were compromised by inadequate models developed to explain system behauiour. This paper outlines the construction of a multi-input multi-output system model which has been used successfully to explain system operation to gas engineers as well as prociding a basis for the design and implementation of a number of advanced control systems. 0 1996 by Elsevier Science Inc.

Keywords: gas transmission, pressure reduction systems, pipeline and valve modelling

1. Introduction

The work described within this paper develops a line of analysis which commenced in the mid-1980s, when research was initiated at British Gas Engineering into system re-inforcement requirements to fulfill the increasing demand for gas. As time progressed, this growth in demand has slowed down significantly, resulting in a change in emphasis from construction towards one of efficient operation.’

The delivery of gas from the national transmission to regional systems and ultimately to the consumer, requires a number of control functions to be performed. The prime objectives when performing these functions are:

1. system safety, integrity, and security of supply; 2. minimizing operation costs; and finally 3. maximizing system throughput and storage.

The performance of gas control systems is influenced by restricted pressure regimes caused by mechanical stress ratings. Up until the 1990s this problem was further exaggerated by redundancies which have to be built into the system because of the poor accuracy achievable using existing pneumatic hardware. In addition, recent increases in demand and seasonal variations have compounded stability and interaction problems.

Modern control technology coupled with major parallel advances in microprocessor technology provided the cata-

Address reprint requests to Prof. C. S. Cox, Department of Engineer-

ing and Advanced Technology, University of Sunderland, Chester Road, Sunderland SRl 35D England.

Received 5 June 1995; revised 16 February 1996; accepted 23 April

1996

Appl. Math. Modelling 1996, Vol. 20, October 0 1996 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010

lyst for the development of schemes to overcome the current operational problems and thereby greatly improve the efficiency of control.* Here one particular section of the transmission system is considered whose role is to supply gas to the regional authorities at required pressures, the above-ground installation (AGI). The AGIs are situated at various locations along the length of the national transmission system pipeline and are configured to extract gas at high pressure for delivery to the regional transmission systems at the required pressure and flowrate. Figure 1 illustrates a typical site configuration where the main regulating stream involves three regulators con- nected in series. The gas enters the stream through a first-stage regulator, which is normally fully open and is activated upon sensing an excess pressure at the station outlet. This is normally referred to as a slamshut, which when activated will provide complete isolation between the input and the output of the station, resulting in the protection of the downstream pipeline from exceeding design pressure limits. Immediately following the slamshut is the first-stage pressure regulating valve, known as the monitor. This operates as a fully modulating regulator in closed-loop pressure control, which is employed to reduce the inlet pressure to some intermediate value between the station inlet and outlet limits. The outlet from the monitor supplies a second regulator referred to as the active, which implements a closed-loop control strategy to regu- late either the outlet pressure or volumetric flow rate. These three regulators form a single stream and the complete structure is called an Active Monitor and Slamshut Strategy. This type of station suffers from the disadvantage that, with certain designs, instability can arise at specific flows and pressure.3 This occurs as a result of interaction between the regulators and connect- ing pipework and manifests itself as a rapidly fluctuating

0307-904X/96/$15.00 PI1 s0307-904x(96)00071-6

Modelling of a two-stage high-pressure gas reduction station: I. Fletcher et al.

PIale - “S YzgEY Figure 1. General layout of an AGI station.

interstage pressure, giving rise to rapid wear of the first- stage regulator (Figure 2).

However, the development of novel control schemes and the analysis of system stability are fraught with many difficulties if attempted directly upon a real plant. Among the problems that may be encountered are excessive cost due to plant inactivity while the identification tests necessary to tune the control schemes and/or study the plant’s behaviour are carried out. This is especially true when attempting to derive transfer function relationships from dynamically slow processes such as the gas transmission system. Further difficulties arise in keeping the system variables constant while attempting to assess the effect of a specific test on the overall system behaviour. In particular, the gas systems considered here will “float” upon the prevailing pressures, whose values differ continuously with demand. Finally the operation of all tests are limited within a specific range determined by the plant’s safety system, which when triggered will significantly change the system characteristics and invalidate the exercise (for ex- ample, the high- and low-pressure overrides in an AGI). Because of these disadvantages, it is often desirable to develop alternative schemes initially using a model of the process.

The word ‘modelling,’ in an engineering context, has two principal meanings. The first (see Section 2) is mathematical modelling, in which the system to be studied is represented by a set of equations developed by applying the appropriate physical (and chemical) laws to each com- ponent. Solution of this set of equations to various sorts of stimuli will then represent the behaviour of the system. These equations, together with the computational means

of solving them, can then be considered as a simulation of the system allowing systematic investigation of control modes or, indeed, the effects of varying any parameter within the jurisdiction of the designer.

The second (Section 3) is associated with physical models where actual hardware is constructed according to appropriate scaling laws such that its behaviour is pre- dictably related to that of the full-scale system. Wind tunnel testing of aircraft and towing tanks for ship design are examples of this particular type of modelling.

2. AGI mathematical model development

The basic structure of AGIs shown in Figure 1 does not illustrate the many different permutations of individual components that occur within the 200 AGIs that exist in practice.

To this end the system was broken down into individual blocks which were then modelled and included within a fixed framework for analysis. The potential benefits of such a study are

I.

II.

III.

2.1

Individual analysis of the elements that comprise the total system would allow considerable simplification of any subsequent stability/controller design study. Information gathered from identification tests upon sections of the real system could be incorporated for evaluation or updating of the model. The effects upon the system of new and enhanced equipment can be investigated prior to its application.

Regulators

The regulating control valves are arguably the most im- portant item of hardware within the AGI station. As a consequence, various sizes and types of regulator are employed within the National Transmission System/Pres- sure Reduction stations, depending upon the policies and politics at the time of construction of the particular au- thority, as well as the predicted loading requirements of the future.

Because the fluid is a gas, inertial effects in the flowing medium can be neglected and the valves characterised by static and dynamic testing.

Figure 2. Illustration of hunting instability in an AGI station. Results from a field test upon an axial flow regulator.

742 Appt. Math. Modelling, 1996, Vol. 20, October


2.2 Steady-state jlow characteristics

Various formulae have been proposed4mh for predicting the steady-state flow rate of gases through the many different types and sizes of commercially available regulator. Most are developed from Bernoulli’s or the continuity equation and differ primarily in the pressure term that is incorporated to describe gas density effects, although some are empirical relationships derived from air testing results.’ Most of these formulae are intended for use over a restricted range of pressure drop conditions, and a more general expression to cover the flow of a compressible medium under isentropic conditions must take account of the transition in a flow regime which occurs at a critical ratio of inlet and outlet pressures where the fluid passing through a controlling aperture reaches sonic velocities. Theoretical treatments of this phenomenon lead to com- plex expressions which are computationally cumbersome.x By introducing sufficient redundancy into the formulation, Buresh and Schuder” have proposed a method that is applicable to virtually all types and sizes of regulator existing today. Utilisation of the universal gas sizing equation within the proposed valve model would permit the use of a fixed structure subroutine, whose predicted output flow would be dependent upon the surrounding system pressure, P, and Pz (psi absolute); temperature T(“R); and the various constants necessary to describe the particular regulator. The basic equation is given by

Q= F .C,C;P,.sin[8]sft”/hr

where

within the limits 0 and f

(1)

where P, and Pz are the upstream and downstream valve pressures, respectively, and C, and C, are coefficients which are dependent upon the valve’s stem position, the first indicating the basic flow capacity of the valve for the critical drop, whereas the latter measures the extent of the pressure recovery of the valve. The inclusion of constants C, and G allows for the various types of gases that can be passed (see Table 1).

Table 1. Gas constants for the universal gas sizing equation

Gas

Specific density

(G)

Correction factor for

specific heat variation

(C,)

Air

Hydrogen

Methane

Natural gas

(Bacton)

Nitrogen

Oxygen

1 .oo 1 .oo

0.07 1 .oo

0.55 0.98

0.60 0.98

0.97 1 .oo

1.10 1 .oo

For liquids, C, is the standard liquid sizing coefficient, which can be determined directly from water tests. How- ever, to determine this coefficient for gases, we must look at the results of air tests when incompressible and compressible flows are equal. That is when

PI - p2 ~ < 0.02

P,

Since under these conditions sin(x) =x, equation (1) can be reduced to

whereas the coefficient C, is dependent upon the critical flow or choked flow (Q,> through the regulator at a particular stem position, that is when sin(x) + 1. Hence

Therefore the size of the valve is expressed in terms of the maximum values of C, and Cg (for 100% opening) and the valve type via the shape of the C, and C, characteristics against stem position, x (see Figure 3). This behaviour is then introduced into the mathematical model using polynomial fits.

2.3 Actuator behaliour

Although many different types of actuator are used within the gas industry, the most prevalent device is the pneumatic actuator. Ideally, the actuator stem travel is propor- tional to the pressure in the actuator chamber. However,

Pwu*g

01

USdM

Fbw

0 10 10 30 40 50 a0 70 00 00 1M

PsmMwm Vah Slam Tmvl

Figure 3. Normalised C, and C, characteristics for a 10”

Fisher V25 regulator.

Appl. Math. Modelling, 1996, Vol. 20, October 743


in practice, its behaviour will also be affected by friction effects, especially at the gland (pressure seal) and by fluid forces acting on the internal components.‘0 To reduce these adverse effects, feedback is commonly used in the form of a valve positioning network.

The speed of response of the valve/converter arrange- ment is limited by the rate at which the actuator chamber can be charged and discharged. To make these rates similar booster relays are often used to permit greater flows into the chamber by working from larger pressures, as shown in Figure 4. This allowing the use of first/second-order transfer functions to describe the almost linear behaviour of the actuation system.

2.4 Pipeline modelling

The dynamics of the system downstream of the second regulator plays a major role in determining the overall stability of the network. It is therefore necessary to ensure that the model used to represent this part of the process in the simulation will reproduce the salient features ade- quately”-‘” without incurring an inordinately high cost in computing time.

Pipelines constitute distributed parameter systems, and a rigorous analysis involves partial differential equations which must be solved under two-point boundary conditions.‘4m” This is, in computer terms, a notoriously time- consuming procedure. We shall, therefore, explore the commonly applied approach of converting the partial differential equations into a set of simultaneous ordinary equations by finite ‘differencing.‘20 In effect, this means discretizing the pipeline into a number of sections, n.‘4X’y,21 It is known that putting n = 1, that is, treating the pipeline as a single fixed volume, oversimplifies the situation to a misleading extent. l4 The first task is, therefore, to find the minimum value for n which will yield acceptable results.

Figure 4. Dynamic response of a Fisher pneumatic actuator

fed by a Fairchild 5200 with booster and positioner. Perfor-

mance is contrasted with a linear first-order model.

744 Appl. Math. Modelling, 1996, Vol. 20, October

One major advantage of this approach is that the resulting mode1 is well suited to the application of standard linear control design techniques.

The nonlinear dynamics of each pipeline element is to be modelled using a resistance/capacitance equivalent network, since in the presence of pipeline wall frictional effects and pressure drops inertial effects may be considered negligible.”

2.5 Pipeline restrictions

The pressure/flow relationship of compressible gas prop- agating down a pipeline is known to exhibit a high degree of nonlinearity. Even in the steady state, numerous relationships have been proposed which evaluate the pressure drop corresponding to a given flow magnitude and direc- tion.2’ Any such eq uation which is to be employed to estimate the discretized pipeline’s pressure profile should incorporate the following:

1. wall frictional forces, which are one of the major con- tributions towards the pressure drop (P, - P,> within a pipeline;‘”

2. the pressure head that occurs due to the geography of the pipeline.

One such equation that satisfies the above criteria, and whose accuracy has been proved in practice” over medium and high pressure ranges, is the genera1 panhandle equation:”

. f .DS

I' (5)

where K’ is a constant,

E, = 0.031055.G.(h,-h,).PAVG?

. *A”G (6)

Z AVG

P,, T, are the base pressure and temperature, respec-

tively; PAvo, zAvo, *AVo are the average pressure, compressibility factor, and temperature of the gas; and L, D are the pipeline dimensions of length and diameter, respectively.

The correction for elevation changes (E,) is based upon average gas density, and the pipeline wall friction factor (f) is determined from a Moody chart using the following equation, which evaluates the system Reynold’s number for natural gas with a specific density (G) of 0.6 (see Table I):

Re = 8880.2 % [ 1 (7)

The friction factor can also be evaluated when under fully turbulent flow conditions via

1 - = [4.1og(3.7 * Relative roughness)? f

(8)


The above equations provide the basis for evaluating the pipeline’s steady-state pressure profile at some specific flow level, under the following assumptions:”

I. The properties of the gas do not vary significantly over a cross section of the pipe.

II. The area of a cross section of pipe is constant. III. The temperature remains constant’s (Figure 5, top)

T = 520”R). IV. The pipe has a relative roughness of 12,500 (f =

0.002872). V. The pipe is horizontal.

Under the above assumptions equation (5) reduces to the following equation, which is used to evaluate pipeline initial conditions for a given flow value:

Pf-P,2= L.Q2

735.5812D5 (9)

All quantities are in SI units.

2.6 Pipeline capacities

To complete the pipeline model, a method of evaluating the pressures at the nodes is required. This was developed from standard thermodynamic theory using the Ideal Gas Law, PV= mRT. Differentiation of this equation with respect to time results in

dP _ =K.k dt

(10)

where k is the mass flow rate, under the following assumptions:

I. Negligible temperature changes II. Constant enthalpy (no external heat added)

III. Constant specific volume, hence constant specific density, its reciprocal

Support for the above assumptions is provided in Figure 5, which illustrates the variations in temperature and density that occurred during the testing of a 4” axial flow valve.‘O The tests were carried out for flow variations of up to 55 m3/sec (7 Msft3/hr), at valve differential pressures rang- ing from 350 KN/m2 (50 psi) to 2400 KN/m2 (350 psi). It is evident from these results that only small changes in system temperature and density have occurred, providing justification of the above assumptions.

Under the constant specific density pi, assumption,

Combining this with the Ideal Gas Law results in13

dP - =K*.Q dt

(11)

4cm c48STellumm lae.iaum

3%.

Tmv 300.

w 2

zm-

Irn-

im-

So-

0 0 1D 20 30 40 so 60

vo)mmc Flw Ral# (mwuc)

@ Qa* LknHy Da4atim

so- MY -

WW 40

Jo-

20.

10.

0 0 10 20 30 40 5a 60

v-s Flow Rd. (mxwc)

Figure 5. Temperature and density variations of a typical pres-

sure reduction station.

where

P,.RT P, K* = pb’K = - = 7 = constant

V

P, being the base pressure and V the pipeline section volume. Note that P, can also be determined as a function of the particular gas flowing.

2.7 Number of stages required?

To satisfy the transient accuracy of the discretized pipeline model an infinite number of stages II are necessary. However, computationally this is impossible because of

Pip&m sinxllum for n=1,2,4&1cI

Figure 6. Evaluation of the optimum number of stages (n)

necessary to reflect pipeline dynamics. Time record shows the

valve throughput of the Winkfield/Ripley telemetry control

scheme to a 20% load flow disturbance.


Modeling of a two-stage high-pressure gas reduction station: I. Fletcher et al.

-1 0 1 2 5 -1 0 1 2 REM REAL

Figure 7. Frequency response of the AGI mathematical model.

the excessive amounts of computer programming required, indicating the need to determine the minimum number of stages which would satisfy the accuracy/computational efficiency balance.

Figure 8. Schematic of the AGI test rig.

This was investigated by using a simulation of the Winkfield-Ripley remote boundary pressure control scheme by comparing the system responses, when the system is experiencing a +20% load flow change, for various values of n. Figure 6 illustrates the responses derived from the above test. From these responses it was found that for four or more stages the differences between the simulations were sufficiently small as to be considered negligible. Further conformation of this result for the ‘optimum’ number of stages was found in a similar study" upon a pipeline model employing continuity equations for the pressure/flow relationships.

By keeping II to a minimum, without significantly de- grading the pipeline’s performance, the order of the system transfer function developed would also be kept to a minimum, ensuring that any study performed upon the AGI simulation would be as uncomplicated as possible.

The trial simulation utilised twin Fisher V25 throttling ball valves, 12”M interconnecting pipework of length 14 diameters, and 3 miles of IS”0 downstream pipeline. The pressure reduction was from a 6900 RN/m2 (100 psiG) supply pressure to 3480 RN/m’ (500 psiG), with 50% of the total drop occurring across each regulator, and a nominal station throughput of 39.3 m3/sec (SMsft”/hr). To develop the transfer functions in terms of the Laplace operator s that describe the behaviour of the simulation at the above operating points the nonlinear simulation was linearised using a small-scale linearisation approach. Figure 7 illustrates the frequency response of these linearised transfer functions in the form of a Nyquist array. Here element i,j refers to the frequency response between the jth input and the ith output with its associated transfer function denoted by Gij(s) (see Table 2). These linearised models have been used extensively in the development of single input-single output (SISO) and multi-input-multi-output (MIMO) control strategies to provide robust control and improved performance.24-2”

2.8 AGI modelling

The AGI network shown in Figure 1 is capable of modelling any of the stations that exist in practice. However, to simplify the simulation and minimise the size of the resulting transfer function the safety systems were omitted and only the normal station operation was considered.

Table 2. AGI multivariable control notation

Input output

1 Monitor’s input Interstage volume

signal (U,) pressure (P,)

2 Active’s input Station inlet

signal (U,) pressure (P,,)

3. Practical evaluation using an experimental test rig

It is well known that the analysis of a nonlinear system is fundamentally difficult for all but the simplest of cases. Features which identify the behaviour of these systems include limit cycles, chaos, catastrophes, switching, and sliding. Any information that can be gathered about the operation of such a system prior to any analysis is of great advantage.

One approach, often favoured by engineers, for obtain- ing such supplementary information about the system is by construction of an experimental test rig, the test rig providing the hardware which can then be used to determine numerical values of the constants and other parameters needed for the system identification. In addition, the



Table 3. AGI test rig hardware

Variable Description Device

Inputs Ul Valve 1 (upstream)

u2 Valve 2 (downstream)

u, Load regulator

JLIN AT0 Platon M-valve (via

Fairchild T5200 I/P)

JLIN AT0 Platon M-valve (via

Fairchild T5200 I/P)

AT0 Platon M-valve (via Fairchild

T5200 l/P)

outputs p, PJY,)

P,“(YZ)

Pd

dPg

P act, P act2

XI

x2

Station inlet pressure

Interstate volume pressure

Station outlet pressure

System downstream load pressure

Station throughput

Valve 1 Actuator pressure

Valve 2 actuator pressure

Valve 1 stem displacement

Valve 2 stem displacement

Sensym LX1 820GE (O/l 00 psiG)

Sensym LX1 820GB (O/l 00 psiG)

Sensym LX1 820GB (O/100 psiG)


Druck PTXllO (O/75 mBar via Dall

tube)



HLP/19O/FLl/l80/4K (O/100 mm)

Penny and Giles

Penny and Giles

HLP/l 9OFLl/l80/4K (O/l 00 mm)

test rig provides a means of verifying the various modelling exercises and allows the results of any analysis to be evaluated using real data collected from a controllable safe environment at minimum cost/time.

3.1 AGI test rig

The test rig (F&-e 8) uses the basic AGI structure as illustrated in Figure I and operates from a supply pressure of 690 KN/m* (100 psiG) and at flow rates up to 100 litres/min in 6 mm (l/4” ) diameter pipework. A solenoid valve is employed as the slamshut, and a pair of Platon M-valves provide monitor and active regulation into a

-0.21 I 0 m 40 60 50 100 im

Figure 9. Normalised open-loop step responses of the AGI test resulting in the parameter in Table 4. rig. Results show both the plant response and its linear model Using MATLAB the above data were used to deter- (detailed below). mine the rig’s Nyquist array (Figure IO). Comparison of

large volume cylinder which helps provide the relatively slow dynamics of the downstream system. Table 3 defines the measurements and transducers/equipment used on the rig, all of which are converted electronically to O/10 V outputs to match standard data acquisition cards. The various measurements taken allow us to investigate regulator performance as well as the various control configura- tions that AGI stations in which can be operated.

3.2 Test rig results

Initial testing commenced using the previously defined pressure control strategy with regulator set points of 75% and 50% of the station inlet pressure. Figure 9 shows the results of open loop step responses collected after the system had been established at the desired steady-state levels with the load regulator 50% open. (Note that this test would take significantly longer on the real system and the data collected would often be corrupted by changes in consumer demand.) The AGI test rig was modelled in the form of the discrete time MIMO transfer function matrix and follows the previous elemental notation:

P” G,Jz) G,Jz) U,

P IN = G&) G&) ’ u, II I

Each element of the matrix G(z) was determined using recursive least-squares identification using the structure (sampling time = 2 sets)

G,j(Z) = b,z-’ + b,zm2

1 -a,z-I -a,z-*



Table 4. Test rig model parameters

Multivariable

G,,(z)

Parameter b, 0.1409

value b, -0.1328

al 0.9530

a2 0.01 14

Transfer Function Element

G,2(~) G,,(z) G,,(z)

-0.1343 0.0013 0.009 1

0.1276 0.0099 0.0075

0.9122 0.4966 0.4957

0.0047 0.4374 0.4277

1 -11

0.5 .___.... i _......_. I__ __....__: . . . . . . . . IMA(3 I

0 n . . . . . . ..I . . . . . . . . . . . . .._.... i . . . . . . . .

_.__..__ j.

/ f 4.5 cr/ ..____. j__. __.__ i . . .._...

1 1 I :

; .,

1 &molt12 j /

j : / j i 1 0.5 _...._..: __“_‘__‘_~““._._..,..‘.“..

i j El 0 _.__.___ + . . . . . . . i . . . . . . . . . L . . . . . . . .

j i 1 4.5 w . . . . . . . . t __._..... f ..__..... ;___...._

j : j j

., : :

REM REM

Figure 10. Frequency response of the AGI test rig

this behaviour with that of the linearised mathematical mode (Figure 7) for the same set point profile indicates the similarities between the behaviour of the two model forms. Indeed the only significant discrepancy is that of the response between movement in the downstream control valve and its effect upon the interstage volume pressure, element G,,. In the mathematical model G,,(s) has pure derivative action, signified by zero steady-state offset in the appropriate open-loop step responses, whereas the test rig response displays phase advance-like behaviour.

4. Conclusion

This paper has explained the development of a detailed mathematical model of a ‘typical’ AGI station which con- tains sufficient structural flexibility to enable it to simu- late any of the 200+ stations that exist in practice.

Validation of the mathematical model was performed using data provided by the Gas Engineering Research Station at Killingworth, experimental tests on individual pieces of hardware, and a specially designed experimental test rig. Results obtained from simulations of the model are presented, and a comparison of these results and those obtained from the AGI test rig shows that the simulations are accurate for the set of test signals considered.

A further indication of the modelling philosophy’s accuracy is provided from the study of remote boundary pressure control systems. 27.28 Here, models developed using the same mathematical relationships employed in the construction of the AGI simulation have been used to design novel control schemes; the resulting controller parameters are directly applicable to the actual systems.

Acknowledgments

The authors wish to express their gratitude to British Gas, SERC, and the University of Sunderland for their support of this work. Particular thanks are extended to the staff of the Engineering Research Station at Killingworth for all the help and guidance they provided.

Nomenclature

ctl CL’, c,

E K’ L A4 P P AVG

Pb

Q Re T T AVG

Tb V Z AVG

correction factor for specific heat variation valve stem position coefficients pipeline diameter friction factor specific density constant pipeline length mass flow rate pressure average pressure of the gas base pressure volumetric flow Reynold’s number temperature average temperature of the gas base temperature pipeline section volume average compressibility factor at the gas

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27876677 gas reduction station model

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