Leak Detection in Gas Pipeline Networks ICSTEP109

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Leak Detection in Gas Pipeline Networks using GLR
Method and Transfer Function based Dynamic Simulation
Model
H. Prashanth Reddy, Shankar Narasimhan, and S. Murty Bhallamudi
Indian Institute of Technology Madras, Chennai-600036
Abstract- The biggest problem with the safe operation of the oil and natural gas
pipelines is development of rupture leaks. Corrosion and pressure surges in
the network cause leaks in the system. Delay in detecting leaks leads to loss of
property and human life in fire hazards and loss of valuable material. Leaking
hydrocarbon gas causes negative impacts on the eco system such as global
warming and air pollution. The objective of this study is to develop a reliable,
sensitive, accurate and computationally fast on-line leak detection method in
complex gas pipeline networks by using state estimation of available pressures
and flow rate measurements, sampled at regular intervals. In this paper we
describe how the GLR method can be adapted for leak detection and
determining the leak location and magnitude in pipeline networks. In this
methodology, to estimate the state of the gas pipeline network, dynamic
simulator based on transfer function approach is combined with data
reconciliation to exploit the redundancy in the measurements. IDEAL gas
model has been implemented to model the state equation of the gas. The above
dynamic model is used repeatedly to test different hypothesis regarding leak
location and leak magnitude. The hypothesis which best fits the data is taken
as right hypothesis. The efficiency of the leak detection methodology with
increase in redundancy in measured data is demonstrated.
Keywords: Dynamic model, gas pipeline networks, transfer function method, data
reconciliation, leak detection, GLR method
I. INTRODUCTION
Leak detection methodologies can be classified into SCADA based continuous leak detection
systems and field investigative leak detection systems. Field detection systems (pigging, acoustic
equipment, tracer gas methods, and infrared photography) do not depend on the available SCADA
measurements but require expensive instruments and skilled personal to monitor the pipeline
system.
Sandberg et al.
1
examined the hydrocarbon sensing cable, which is constructed of an alarm signal,
continuity wire and two sensor wires installed along the pipeline, to monitor pipelines for small or
large hydrocarbon solvents leakages. Watanabe and Himmelbalu
2
developed acoustic method
based on impulse response of the acoustic wave in the pipeline which has a sharp pulse at a certain
time can be directly related to the site of the leak. Pudar and Liggett
3
formulated an inverse leak
detection method for water distribution networks with equivalent orifice areas of positive leaks as
unknowns. Billmann and Isermann
4
proposed a leak detection method based on mathematical
dynamic models, non linear adaptive state observers and a correlation detection technique.
Mukherjee and Narasimhan
5
employed generalized likelihood ratio method in combination with
Proceedings of International conference on Sustainable technologies for Environmental Protection- ICSTEP2006
1
modified serial compensation strategy to detect, locate and estimate multiple leaks in water
distribution networks. Belsito et al.
6
developed a leak detection system for liquefied gas pipelines
using artificial neural networks (ANN) for leak sizing and location by processing the field data.
From a critical analysis of the state of the art leak detection methods it is noted that none of the
methods is a universal solution for the leak detection in gas pipeline networks. The objective of
this study is to develop a reliable, sensitive, accurate and computationally fast on-line leak
detection method in complex gas pipeline networks by using state estimation of available pressures
and flow rate measurements, sampled at regular intervals. In this methodology, to estimate the state
of the gas pipeline network, dynamic simulator based on transfer function approach is combined
with data reconciliation to exploit the redundancy in the measurements. The performance of GLR
method combined with dynamic simulator for detecting leak (in terms of accuracy of location and
magnitude of the leak) is studied through realistic pipeline.
METHODOLOGY OF DYNAMIC SIMULATION BY TRANSFER FUNCTION MODEL
A. Governing equations for a pipeline
The governing one-dimensional hyperbolic partial differential equations describing the
conservation of mass and momentum for the unsteady subsonic flow of a gas through a constant
diameter, rigid pipe are
7,8
:
Continuity Equation:
0 ) / (
2
·


+


t
p
c A
x
M
(1)
Momentum Equation:
0
1
) 2 /( / ) sin (
2 2 2
·


+ + +


t
M
A
p DA M M c c p g
x
p
λ θ (2)
Equation of State:
2
/ c zRT p · · ρ (3)
where p is pressure; M is mass flow rate; g is acceleration due to gravity; ρ is density; θ is
inclination of the pipe; λ is coefficient of friction; D is inner diameter of the pipe; A is cross
sectional area of the pipe; c is the pressure wave velocity; x is distance along the pipeline; and t is
time. Compressibility factor, z is a function of pressure and temperature for a given gas. In this
work IDEAL gas law (z = 1) is used to model the compressibility factor of the Natural gas. The
friction factor λ is determined using the Haaland
7
explicit equation.
Proceedings of International conference on Sustainable technologies for Environmental Protection- ICSTEP2006
2
1.11
10
1 6.9
1.8log
3.7
e
D
R
ε
λ
1
¸ _
1

·− +
1

¸ , 1
¸ ]
(4)
where ε is the roughness height of the pipe; and R
e
is the Reynolds number.
A. Transfer Function Model
The transfer function model of a pipe relates the upstream pressure and mass flow rate to the
downstream pressure and mass flow rate in the Laplace domain. To transform the state space
model to Laplace domain, equations (1) and (2) are linearized in terms of deviation variables from
steady state solution (ΔM = M-M
0,
ΔP = P-P
0
) around steady state flow rate and average steady
state pressure in the pipeline. Laplace transforms are applied to linearized PDEs to convert them
into linear differential equations which are then solved analytically to obtain the relations between
the flows and pressures at the ends of the pipe in the Laplace domain
7, 8
.
1 1 , 1 2 2 , 1 1
P F M F M
P M M M
∆ + ∆ · ∆
(5)
2 2 , 2 1 1 , 2 2
M F P F P
M P P P
∆ + ∆ · ∆
(6)
where subscripts “1” and “2” denote upstream and downstream ends of a pipe, respectively. The
transfer functions are approximated by first order series expansion of hyperbolic functions in Eq.
(5) and (6) and can be expressed as
7, 8
Ts
k F
P P
+
·
1
1
1 1 , 2
;
Ts
s T
F
P M
+
·
1
1
1 , 1
;
Ts
F
M M
+
·
1
1
2 , 1
;
Ts
s T
k F
M P
+
+
− ·
1
) 1 (
2
2 2 , 2
(7)
where
2 2
sin
2
c
gL
c
u u
D
L θ λ
− · Ψ
;
Ψ
· e k
1
;
/ 2 2
2
1
1
24
L
k e u
DA
λ
Ψ
¸ _
· + Ψ

¸ ,
;
/ 2 2
1 2
1
1
24
AL
T e
c
Ψ
¸ _
· + Ψ

¸ ,
;
2
2 2 2
1 1
6
1
24
L u
D
T
u
Dc
λ
λ
¸ _


· +

¸ _ Ψ
+



¸ , ¸ ,
;
2
/ 2 2
2
1 1
1
6 24
2
L u
T e
Dc
λ
Ψ
¸ _
· − Ψ + Ψ

¸ ,
; T R z c ·
2
;
2
0 0 0
/( ) u M c p A ·
Our objective here is to use the above simplified model to solve the state estimation problem in the
time domain. Therefore, equations (7) are substituted in Eqs. (5) and (6), which are then
analytically inverted. For this purpose, the time domain functions for ΔP s and ΔM s are first
expressed using their discrete values at regularly spaced time intervals, T
s
. The convolution
Proceedings of International conference on Sustainable technologies for Environmental Protection- ICSTEP2006
3
theorem is then applied to obtain the Laplace inverse of Eqs. (5) and (6) to obtain the following
equations in the time domain
7
.
( ) ( )
2 1 1 2 2
1 1
( )
2 2 2 2
2 2
1
( ) (1 ) ( ) (1 ) ( )
- (1 ) ( ) ( )
s s s s
s s
T T T T
N N
N i N i
T T T T
s s s
i i
T T
N
N i
T T
s s
i
p NT k e e p iT k e e M iT
k T k T
e M iT e M NT
T T
− − − − − −
· ·
− − −
·
1 1
∆ · − ∆ + − − ∆
1 1
¸ ] ¸ ]
1
1

− ∆ + ∆
1
1
1
¸ ]
¸ ]
∑ ∑

(8)
( ) ( )
1
1 2 1
1 1
( ) (1 ) ( ) - (1 ) ( )

s s s s
T T T T
N N
N i N i
T T T T
s s s
i i
T
M NT e e M iT e p iT e
T
− − − − − −
· ·
1 1 1
∆ · − ∆ + − ∆
1 1 1
1
¸ ] ¸ ] ¸ ]
∑ ∑
1
1
+ ( )
s
T
p NT
T

(9)
where, current time t = N × T
s
. It is to be noted that the summation terms on the right hand side of
the above equations are weighted sums of the discrete values of pressures and flows at all times
from the initial to the current time.
The complete discrete model in the time domain for the entire network is obtained by combining
Eqs. (8) and (9) for all the pipe elements with (i) continuity equation and pressure equilibrium
equations at the junctions, (ii) continuity equation and pressure drop equation at valves, (iii)
continuity equation and equations describing compressor operation, and (iv) the boundary
conditions.
B. Formulation of the State Estimation Problem
Eq. (8), (9) for all the pipes and the compatibility equations at the junctions for a network
consisting of only pipes result in a linear system of equations. The linear system of equations is
solved in state estimation frame work. In the state estimation problem, the adopted definition of the
best estimate of the state is that which minimizes the quadratic difference between the measured
values and the estimated values. To make state estimation problem computationally faster for
online implementation, the state estimate for the current time instant NT
s
is obtained in recursive
manner by utilizing the estimates obtained for all the previous times. Therefore at the current time
instant NT
s
the linear system of equations
7
become
Am Bu c + · (10)
where
, m u
are the measured and unmeasured variables corresponding to the current time instant
NT
s
. Vector c contains the weighted sum of the estimated flows and pressures at the previous times
((N-n)T to NT, where n = number of past data samples required). The matrices A and B depend on
the pipe parameters, sampling period, compressibility factor and friction factor. Matrices A and B
are time dependent because the compressibility and friction factor vary with pressure, which is
time dependent.
Proceedings of International conference on Sustainable technologies for Environmental Protection- ICSTEP2006
4
In the state estimation context, the best estimates of variables m and u, for a given set of noisy
measurements y for the variables m, for the current time instant NT
s
can be obtained by solving the
following weighted least squares estimation problem.
1
m,u
Min ( ) ( )


T
y m Q y m
Subjected to
Am Bu c

− −
+ ·
(11)
where the matrix Q is the covariance matrix of errors in measurement. The above problem is a
standard linear estimation (reconciliation) problem for which the solution can be obtained by
Crowe’s Projection Matrix technique as discussed by Narasimhan and Jordache
9
.
If just enough variables are specified, then the solution to the above problem will correspond to an
objective function value of zero. This would correspond to a simulation problem. If more
measurements (specifications) than the minimum required to solve the problem are given, then the
above formulation gives a best fit solution, taking into account the inaccuracies in the measured
data.
Leak Detection and Isolation
In order to detect a leak online, we will use a moving window approach. At each time t, we will
solve Eq. (11) based on the window of measurements [t, t-nT]. If the objective function value
(which represents the goodness of fit) is large, then we can conclude that a leak may have
occurred. It can be shown that under the assumption that the measurement errors follow a
Gaussian distribution; the objective function value follows a chi-square distribution with degrees
of freedom equal to m-u. Where m is number of equations available and u is number of
unmeasured variables. Thus, we can choose the criterion from the chi-square distribution for a
given level of significance in order to decide whether the objective function value is statistically
significant.
If a leak is detected at time t, we will collect measurements for a time period [t, t+NT] where N is a
user defined delay and use this set of measurements to identify the leak branch, location and
estimate its magnitude as follows.
A leak is assumed in every branch of the pipeline network in turn and the best fit leak location in
that branch and best fit magnitude are determined based on the measurements [t, t+NT]. The
hypothesis that best fits the data then determines the branch, location and magnitude of the leak.
The best fit leak location and magnitude for each branch requires an optimization problem to be
solved. The formulation and solution of the optimization problem is described below.
Let us hypothesize a leak in branch i of the pipeline network. Let us assume a location x
l
in the
pipe at which the leak is present. The simplified model for the pipeline network under this
assumption can be constructed as follows. For all pipes j≠ i the equations will be assembled as in
Eq. (10), that is, the coefficients of matrix A, B in the rows corresponding to the flow and
momentum balance equations will be computed as before. At assumed leak location the branch is
Proceedings of International conference on Sustainable technologies for Environmental Protection- ICSTEP2006
5
P
1
, M
1
P
2
, M
2
D
L
= b
L
(unknown leak magnitude)
P
L
, M
L
Figure 2: Branch i divided into two pipes by creating an extra node at the leak
location
x
l
L-x
l
b
L
(unknown leak magnitude)
x
l
L-x
l
P
1
, M
1
Figure 1: Branch i with leak of magnitude b
L
at position x
l
P
2
, M
2
divided into two pipes and an extra node is created with an unknown demand D
L
which is equal to
leak magnitude. One extra node and one extra pipe element are added to the size of the problem.
Therefore existing set of equations are increased by four equations. Two additional equations
required for the extra pipe element and two additional equations are required for the extra node.
Refer Fig. 1 and Fig. 2.
The equations are assembled in the form of Eq. (10) for every pipe and include the mass balance
and pressure equality constraints at each branch node. These set of equations are then solved as
weighted least square state estimation problem (Eq. 11). The unknown D
L
is part of the B matrix.
In order to find the best fit solution for the leak location, we need to find the optimal value of x
l
by
using an univariate golden section approach. This implies that for each guess value of the leak
location, the matrices A and B have to be computed and Eq. (11) has to be solved. The least
objective function value corresponding to the best fit estimate of the leak location is stored along
with the estimates for m and x
l
. We repeat this procedure for each pipe (each hypothesis) in turn
and determine the least objective function value among all the hypotheses, and identify a leak in
the network. The magnitude of the leak can now be computed by using the estimates for m and x
l
which involves back-solving for u and b
l
using Eq. (10).
RESULTS AND DISCUSSION
Accuracy and applicability of the proposed approach is evaluated through simulations on a
hypothetical pipeline shown in figure 3. The pipeline has a uniform circular cross section area of
internal diameter 0.428m throughout. Internal roughness height of the pipeline considered for
friction loss computations is 250 microns. The pipeline consists of 8 full bore sectionalized ball
valves. Natural gas (composition given) having a viscosity of 0.0000125 N.s/m
2
is considered in
the simulations. Compressibility factor is considered as 1.0 (IDEAL gas) in simulations. A
Proceedings of International conference on Sustainable technologies for Environmental Protection- ICSTEP2006
6
constant ambient temperature of 300K is specified at all points in the network. In all the leak
detection runs 10 sec sampling interval (computational time step) is considered. The measurements
required for various leak simulations presented in this paper are generated by gas dynamic model
based on the transfer function methodology
13
. This is considered as forward problem. In leak
detection runs transfer function based gas dynamic model is used with generalized likelihood ratio
(GLR) method. This is considered as inverse problem.
The given 200.7 km pipeline (Figure 3) is divided into 17 segments. Maximum length of a pipe
segment in this case is 15.25 km. It is assumed that mass flow rate and pressure measurements at
source node, demand and pressure measurements at consumption node and intermediate pressure
measurements at sectionalized valves SV-1, SV-4, SV-5 and SV-7 are available. This is the basic
instrumentation level considered in leak detection runs however in some runs, to study
improvement in the leak detection ability of the proposed method with noisy measurements, a
number of intermediate pressure measurements are added to the existing level of instrumentation
as explained below.
SV-2
SV-3 SV-4 SV-5
SV-6 SV-7 SV-8
Power
plant
SV-1
18" pipeline
CH. 017.50 KM CH. 019.565 KM
CH. 037.40 KM
CH. 061.898 KM
CH. 086.2 KM
CH. 111.493 KM CH. 139.536 KM CH. 167.917 KM
CH. 200.7 KM
Figure 3: hypothetical network
All the leak detection simulations are carried out in conjunction with a transient state prevailing
due to a demand variation at the consumption node. Normal demand at consumption node is 1.45
MMSCMD. In all the test cases constant source pressure is taken as 41.62 kg/cm
2
. To create
transient in the pipeline, the demand at consumption node is increased linearly from its original
value by 10% in 25 sampling intervals, kept constant for 25 sampling intervals, and subsequently
brought back to its original value in another 25 sampling instants. Leak is created during the
increasing phase of the demand at consumption node. This is done to test the ability of the
proposed method to isolate the leak during transient conditions in the pipeline.
In this study leak detection tests are carried out under four categories. In category 1, studies are
carried out with existing instrumentation but without noise in the measurements. For online
applications, measurement noise is unavoidable. Therefore, in remaining leak detection runs in
other categories, noise in the flow rate (0.5% of 1.45MMSCMD) and noise in the pressure
(0.075% of 41.62 kg/cm
2
) are considered. In category 2, leak detection runs are similar to category
Proceedings of International conference on Sustainable technologies for Environmental Protection- ICSTEP2006
7
1 runs but noise in incorporated in the measurements. It is expected that when noise is present in
the measurements then accuracy of the leak detection method decreases. Therefore, in category 3
leak detection simulations are similar to category 2 simulations, but a filter is used to reduce the
noise in the measurements. In category 4 runs, extra pressure measurements are added to existing
instrumentation (noise is not filtered out) in order to improve the leak detection accuracy.
Category 1:
Leak simulations under first category are carried out to study the ability of the proposed method to
detect and isolate even small size leaks. To test the consistency of the methodology, leak
magnitudes of 2%, 5% and 10% (percent of total flow rate through the pipeline) at three locations
are considered in these tests. Distances from the source node to Location-1, Location-2 and
Location-3 are 28.2 km, 115.5 km and 170.2 km, respectively. Results from these runs are
summarized in table 1.
Table: 1. Results of leak detection tests without noise in the measurements and with only
existing instrumentation
S. No Leak location
(km)
Magnitude of
leak tested (% of
total flow)
% error in
estimated
magnitude
from actual
magnitude
Error in
estimated leak
location from
actual leak
location (km)
Delay in leak
detection time
from actual leak
time (sec)
1 28.2 2 0.01 0.16 20
2 28.2 5 0.03 0.10 10
3 28.2 10 0.1028 0.11 10
4 115.5 2 6.85 0.17 60
5 115.5 5 0.11 0.17 40
6 115.5 10 1.41 0.17 30
7 170.2 2 1.39 0.31 80
8 170.2 5 6.55 0.21 40
9 170.2 10 0.78 0.21 40
Results presented in the table 1 shows the ability of the proposed method to detect and isolate leaks
with less than 7% error in leak magnitude estimate and approximately 300 m error in leak location
estimate.
Category 2:
Ability of the proposed method to estimate leak location and leak magnitude, when measurement
noise is present, is demonstrated through these runs. Only existing level of instrumentation is
considered. Tests are carried out at location-2 with 2%, 5% and 10% leak magnitude. Results are
presented in Table 2. Results in Table 2 show that error in leak detection and isolation has
increased considerably because of presence of measurement noise.
Table: 2. Results of leak detection tests with noise in the measurements and with only existing
instrumentation
Proceedings of International conference on Sustainable technologies for Environmental Protection- ICSTEP2006
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S. No Leak location
(km)
Magnitude of
leak tested (% of
total flow)
% error in
estimated
magnitude
from actual
magnitude
Error in
estimated leak
location from
actual leak
location (km)
Delay in leak
detection time
from actual leak
time (sec)
1 115.5 2 68.83 14.9 700
2 115.5 5 820.9 1.5 590
3 115.5 10 37.5 1.2 280
Category 3:
In these runs, an exponential moving average filter is used to reduce the measurement noise.
Reduction in measurement noise can be controlled by varying filter constants. The weightages
given to the previous measurements and the current measurements are 90% and 10%, respectively.
In these runs, only existing level of instrumentation is used. Table 3 presents the summary of these
studies. It can be seen that error in leak detection and isolation has decreased significantly when
an exponential moving average filter is used to reduce the measurement noise.
Table: 3. Effectiveness of the Filter in improving the leak detection and isolation abilities
with only existing instrumentation
S. No Leak location
(km)
Magnitude of
leak tested (% of
total flow)
% error in
estimated
magnitude
from actual
magnitude
Error in
estimated leak
location from
actual leak
location (km)
Delay in leak
detection time
from actual leak
time (sec)
1 115.5 2 27.8 2.96 540
2 115.5 5 3.715 1.19 280
3 115.5 10 25.08 0.18 230
Category 4:
In this category leak detection ability of the proposed method is studied with increased redundancy
in measurements to reduce the effect of noise. In these leak detection runs, pressure measurements
are made available at every node i.e. in addition to the existing measurements considered in other
studies, twelve more pressure measurements are available. In these runs 2%, 5% and 10% leaks are
tested at location -2. Results are presented in table 4. It is seen from table 4 that increased
redundancy in measurements improves the accuracy of the leak detection method.
Table: 4. Improvement in the leak detection results with increased redundancy in
measurements to reduce the noise in the measurements
S. No Leak location
(km)
Magnitude of
leak tested (% of
total flow)
% error in
estimated
magnitude
from actual
magnitude
Error in
estimated leak
location from
actual leak
location (m)
Delay in leak
detection time
from actual leak
time (sec)
1 115.5 2 33.8 1.71 290
2 115.5 5 3.0 0.55 100
3 115.5 10 3.0 0.29 40
II. CONCLUSIONS
Ability of the proposed method to accurately detect and isolate leak sizes of 2%, 5% and 10% of
normal flow rate is tested for a 200.7 km pipeline. Leaks are assumed to occur during unsteady
Proceedings of International conference on Sustainable technologies for Environmental Protection- ICSTEP2006
9
flow conditions (due to demand variation) in the pipeline. Accuracy of the proposed method
decreases when measurement noise is present. This is especially true when the leak magnitude is
small. It is demonstrated that improvement in the leak detection accuracy can be achieved either by
using a filter to reduce the measurement noise or by increasing the redundancy in the
measurements. It is shown that increasing the redundancy in the measurements is a better option
than using the filters.
ACKNOWLEDGEMENT
This research work was financially supported by GAIL (India) Ltd. under sponsored project
“Development of leak detection methods in gas pipeline networks”.
VI. REFERENCES
1. Sandberg, C.; Holmes, J.; McCoy, K.; Koppitsch, H.; “Application of a continuous leak
detection system to pipelines and associated equipment”, IEEE Trnas. Ind. Applic., 1989, Vol.
25, pp. 906-909.
2. Watanabe, K.; Himmelblau, D. M., “Detection and location of a leak in a gas-transport pipeline
by a new acoustic method’, 1980, AIChE Jl.. Vol. 32(10), 1690-1701.
3. Pudar, R. S.; Liggett, J. A. “Leaks in pipe networks”, Journal of Hydraulic Engineering,
ASCE, July, 1992, Vol. 118(7), pp. 1031-1046.
4. Billmann, L.; Isermann, R.; “Leak Detection Methods for Pipelines”, Automatica, 1987, Vol.
239(3), pp. 381-385.
5. Mukherjee, J.; Narasimhan, S., “Leak detection in networks of pipelines by the generalized
likelihood ratio method”, Ind. Eng. Chem. Res, 1996, Vol. 35(6), 1886-1893.
6. Belsito, S.; Lombardi, P.; Andreussi, P.; Banerjee, S. “Leak detection in liquefied gas pipelines
by artificial neural networks”, AIChE Journal, December 1998, Vol. 44(12), pp. 2675-2688
7. Reddy, H., P.; Bhallamudi, S., M.; Narasimhan, S., “Simulation and state estimation of
transient flow in gas pipeline networks using transfer function model”, Ind. Eng. Chem. Res.
(under review).
8. Kralik, J.; Stiegler, P.; Vostry, Z.; Zavorka, J. “Modeling the dynamics of flow in gas
pipeline,” IEEE Trans. Syst., Man, Cybern., July/August 1984, Vol. SMC-14, No. 4
9. Narasimhan, S.; Jordache, C. Data reconciliation & Gross Error Detection – An Intelligent
Use of Process Data, Gulf Publishing Company, Houston, Texas, 2000
Proceedings of International conference on Sustainable technologies for Environmental Protection- ICSTEP2006
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