Leak Detection in Gas Pipeline Networks ICSTEP109

Leak Detection in Gas Pipeline Networks using GLR Method and Transfer Function based Dynamic Simulation

ModelH. 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. INTRODUCTIONLeak 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 Himmelbalu2 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 Isermann4 proposed a leak detection method based on mathematical dynamic models, non linear adaptive state observers and a correlation detection

Proceedings of International conference on Sustainable technologies for Environmental Protection- ICSTEP2006

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technique. Mukherjee and Narasimhan5 employed generalized likelihood ratio method in combination with 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.

II. METHODOLOGY OF DYNAMIC SIMULATION BY TRANSFER FUNCTION MODEL

A. Governing equations for a pipelineThe 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 are7,8 :

Continuity Equation:

(1)

Momentum Equation:

(2)

Equation of State:

(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 is determined using the Haaland7 explicit equation.


2

(4)

where is the roughness height of the pipe; and Re is the Reynolds number.

A. Transfer Function ModelThe 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-M0, ΔP = P-P0) 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 domain7, 8.

(5)

(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 as7, 8

; ; ; (7)

where

; ; ; ;

; ; ;

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, Ts. The convolution


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theorem is then applied to obtain the Laplace inverse of Eqs. (5) and (6) to obtain the following equations in the time domain7.

(8)

(9)

where, current time t = N × Ts. 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 ProblemEq. (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 NTs is obtained in recursive manner by utilizing the estimates obtained for all the previous times. Therefore at the current time instant NTs the linear system of equations7 become

(10)

where are the measured and unmeasured variables corresponding to the current time instant NTs. 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.


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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 NTs can be obtained by solving the following weighted least squares estimation problem.

(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 Jordache9.

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.

C. 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 xl 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 divided into two pipes and an extra node is created with an unknown demand DL which is


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P1, M1 P2, M2

DL= bL (unknown leak magnitude)

PL, ML

Figure 2: Branch i divided into two pipes by creating an extra node at the leak location

xl L-xl

bL (unknown leak magnitude)

xl L-xl

P1, M1

Figure 1: Branch i with leak of magnitude bL at position xl

P2, M2

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 DL 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 xl

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 xl. 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 xl which involves back-solving for u and bl using Eq. (10).

III. 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/m2 is considered in the simulations. Compressibility factor is considered as 1.0 (IDEAL gas) in simulations. A


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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 methodology13. 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.

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/cm2. 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/cm2) are considered. In category 2, leak detection runs are similar to


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category 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 0.16 202 28.2 5 0.03 0.10 103 28.2 10 0.1028 0.11 104 115.5 2 6.85 0.17 605 115.5 5 0.11 0.17 406 115.5 10 1.41 0.17 307 170.2 2 1.39 0.31 808 170.2 5 6.55 0.21 409 170.2 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.


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Table: 2. Results of leak detection tests with noise in the measurements and with only existing instrumentation



total flow)






1 115.5 2 68.83 14.9 7002 115.5 5 820.9 1.5 5903 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



total flow)






1 115.5 2 27.8 2.96 5402 115.5 5 3.715 1.19 2803 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



total flow)



actual leak location (m)



1 115.5 2 33.8 1.71 2902 115.5 5 3.0 0.55 1003 115.5 10 3.0 0.29 40


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IV. 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 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.

V. ACKNOWLEDGEMENT

This research work was financially supported by GAIL (India) Ltd. under sponsored project “Development of leak detection methods in gas pipeline networks”.

VI. REFERENCES1. 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


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Leak Detection in Gas Pipeline Networks ICSTEP109

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