OBSERVER DESIGN USING BOUNDARY INJECTIONS FOR ...
ADCHEM 2006 International Symposium on Advanced Control of Chemical Processes Gramado, Brazil – April 2-5, 2006
OBSERVER DESIGN USING BOUNDARY INJECTIONS FOR PIPELINE MONITORING AND LEAK DETECTION Ole Morten Aamo, Jørgen Salvesen, and Bjarne A. Foss
Department of Engineering Cybernetics Norwegian University of Science and Technology N-7491 Trondheim
Abstract: We design a leak detection system consisting of an adaptive Luenbergertype observer based on a set of two coupled one dimensional first order nonlinear hyperbolic partial di erential equations governing the flow dynamics. It is assumed that measurements are only available at the inlet and outlet of the pipe, and output injection is applied in the form of boundary conditions. For the linearized model without friction and leak, exponential convergence of the state estimates to the plant state is shown by Lyapunovs method. The observer design is performed for the continuum model, ensuring that convergence properties established theoretically are not an artifact of the method of discretization. Heuristic update laws for adaptation of the friction coe!cient and leak parameters are given, and simulations demonstrate their ability to detect, quantify and locate leaks. Keywords: Partial di erential equations; Observers; Adaptive systems; Pipeline leaks
1. INTRODUCTION
box), while others incorporate models based on physical principles. Our method falls into the latter category, in that we will use a dynamic model of the pipe flow based on a set of two coupled hyperbolic partial di erential equations.
Transportation of liquids in pipelines requires monitoring to detect malfunctioning such as leaks. In the petroleum industry, leaks from pipelines may potentially cause environmental damage, as well as economic loss. These are motivating factors, along with requirements from environmental authorities, for developing e!cient leak detection systems. While some leak detection methods are hardware-based, relying on physical equipment being installed along the pipeline, the focus of this paper is on software-based methods that work for cases with limited instrumentation. In fact, instrumentation in the petroleum industry is usually limited to the inlet and outlet of pipelines, only. This calls for sophisticated signal processing methods to obtain reliable detection of leaks. Some software-based leak detection methods perform statistical analysis on measurements (black
IFAC
There have been numerous studies on model based leak detection. We mention here the most relevant ones with regard to our work. Based on a discretized pipe flow model, Billman and Isermann (1987) designed an observer with friction adaptation. In the event of a leak, the outputs from the observer di ers from the measurements, and this is exploited in a correlation technique that detects, quantifies and locates the leak. Verde (2001) used a bank of observers, computed by the method for fault detection and isolation developed by Hou and Müller (1994). The underlying model is a linearized, discretized pipe flow model on a grid of Q nodes. The observers are designed in
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ADCHEM 2006
such a way that all but one will react to a leak. Which one of the Q observers that does not react to the leak depends on the position of the leak, and this is the mechanism by which the leak is located. The outputs of the remaining observers are used for quantifying the leak. The bank of observers are computed using the recursive numerical procedure suggested by Hou and Müller (1994), however it was shown in Salvesen (2005) that due to the simple structure of the discretized model, the observers may be written explicitly. This is important, because it removes the need for recomputing the bank of observers when the operating point of the pipeline is changed. Verde (2004) also proposed a nonlinear version, using an extremely coarse discretization grid.
and the momentum conservation (ignoring friction for now) Cx 1 Cs Cx +x + = 0> Cw C{ C{
(2)
for ({> w) 5 (0> O) × (0> 4), and where x is flow velocity, s is pressure, and is density. The relation between pressure and density is modelled as (Nieckele et al. (2001)) s suhi > (3) f2 where uhi is a reference density at reference pressure suhi , and f is the speed of sound. Equation (1)—(2) also describes gas flow in a pipe, simply by replacing (3) with the ideal gas law. Under the conditions we consider, we assume f is su!ciently large to ensure A 0= In compact form, we have ¸ ¸ C s C s + D (s> x) = 0> (4) Cw x C{ x = uhi +
Several companies o er commercial solutions to pipeline monitoring with leak detection. Fantoft (2005) uses a transient model approach in conjunction with the commercial pipeline simulator OLGA2000, while EFA Technologies (1987, 1990, 1991) uses an event detection method that looks for signatures of no-leak to leak transitions in the measurements.
where, using (3), 5
6 x n+s 8> D (s> x) = 7 f2 x n+s n = f2 uhi suhi >
The detection method of Verde (2001) using a bank of observers, can potentially detect multiple leaks. However, multiple simultaneous leaks is an unlikely event, so the complex structure of a bank of Q observers seems unnecessary. In this paper, we instead employ ideas from adaptive control, treating the size and location of a single point leak as constant unknown parameters. This differs from the method of Billman and Isermann (1987), since we will model the leak in the observer, thereby obtaining state estimates also in a leak situation. Another important aspect of our method is that the observer is designed for the continuum model, ensuring that convergence properties established theoretically are not an artifact of the method of discretization. Our leak detection system consists of an adaptive Luenbergertype observer, based on a set of two coupled one dimensional first order nonlinear hyperbolic partial di erential equations governing the flow dynamics. It is assumed that measurements are only available at the inlet and outlet of the pipe, and output injection is applied in the form of boundary conditions. Heuristic update laws for adaptation of the friction coe!cient and the two leak parameters are suggested.
(5) (6)
and the boundary conditions are x (0> w) = x0 (w) >
(7)
s (O> w) = sO (w) =
(8)
The eigenvalues of D are 1 = x f> 2 = x + f=
(9)
Assuming that x ¿ f> which is always the case in the applications we are considering, the eigenvalues are distinct and satisfy 1 ? 0 ? 2 =
(10)
The system is therefore strictly hyperbolic. Steady state solutions (¯ s> x ¯) of (4), must satisfy ¸ C s = 0= (11) D (s> x) C{ x Since D (s> x) is invertible (l 6= 0), we have that Cx ¯ C s¯ = 0> = 0> (12) C{ C{ so s¯ and x ¯ are constant. The boundary conditions ¯ = x0 . (7)—(8) yield s¯ = sO and x
2. MATHEMATICAL MODEL 2.2 Model in Characteristic Form 2.1 Physical Model Consider now the change of coordinates
For liquid flow in a pipe we have the mass conservation Cs Cx Cs +x + f2 = 0> (1) Cw C{ C{
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µ (s> x) = f ln
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¶ n+s +xx ¯> n + s¯
(13)
ADCHEM 2006
µ (s> x) = f ln
¶ n+s +xx ¯> n + s¯
sˆ (O> w) = sO (w) =
(14)
Notice that the input to (4)—(8) is also copied in (25)—(27). Equation (25)—(27) can be viewed as a Luenberger-type observer, and convergence is guaranteed when the process is operated at asymptotically stable fixed points, which is usually the case for pipelines. However, we look for alternatives to the boundary conditions (26)—(27) which yield better convergence properties. Taking s0 (w) = s (0> w) and xO (w) = x (O> w) as process measurements, we may apply output injection to (25)—(27). In transformed coordinates, we obtain
which clearly is defined for all physically feasible s and x. It is easy to see that it’s inverse is s (> ) = (n + s¯) exp (( ) @ (2f)) n> (15) x (> ) = x ¯ + ( + ) @2=
(16)
Notice that the fixed point (¯ s> x ¯) corresponds to (0> 0) in the new coordinates. The time derivative of (13)—(14) is sw + xw > n+s sw + xw = w = f n+s
w = f
³ ³ ´ ´ ˆ @2 ¯+f+ ˆ + ˆ { = 0> ˆw + x ³ ³ ´ ´ ˆ @2 ˆ = 0> ˆ + x ¯f+ ˆ + w {
(17) (18)
Inserting for sw and xw from (4) yields µ
xs{ + (n + s) x{ f2 s{ + xx{ n+s n+s ¶ µ s{ + x{ = (x + f) f n+s = (x + f) { > (19) µ 2 ¶ xs{ + (n + s) x{ f s{ + xx{ w = f n+s n+s ¶ µ s{ + x{ = (x f) f n+s = (x f) { = (20)
w = f
x f + ( + ) @2) { = 0= w + (¯
(22)
where e0 and eO are functions to be designed. Notice that the boundary injections (30)—(31) may be any function of the known signals at both ends of the pipe. For convergence analysis, we consider the linearization of (21)—(22) and (28)— (29) around (0> 0) and form the dynamics of the ˜ = = ˆ observer error, defined as ˜ = ˆ, We obtain ¸ ¸ C C ˜ ˜ (32) ˜ + C{ ˜ = 0> Cw
The boundary conditions are obtained from (15)— (16), and are (0> w) + (0> w) = 0>
(23)
(O> w) (O> w) = 0=
(24)
with boundary conditions ³ ˆ (0) > ˜ (0) = e0 (0) > (0) > ˆ (0) > ˜ (0) + ´ ˆ (O) > (33) (O) > (O) > ˆ (O) >
The characteristic form (21)—(22) is convenient for the observer design carried out in the next section.
³ ˆ (0) > ˜ (O) = eO (0) > (0) > ˜ (O) ˆ (0) > ´ ˆ (O) > (34) (O) > (O) > ˆ (O) >
3. OBSERVER DESIGN In reality, input signals to pipelines are usually choke openings at the inlet and outlet. Here, we instead view x0 (w) and sO (w) in (7)—(8) as inputs to the process, and construct the copy of the plant dynamics (4) ¸ ¸ C sˆ C sˆ + D (ˆ s> x ˆ) = 0> (25) ˆ ˆ Cw x C{ x
where
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=
¸ x ¯+f 0 = 0 x ¯f
(35)
Following Xu and Sallet (2002) and Coron et al. (2004), consider the Lyapunov function candidate Z O 1 ˜ 2 h{@(f+¯x) g{ Y = f+x ¯ 0 Z O 1 ˜ 2 h{@(f¯x) g{> (36) + fx ¯ 0
with boundary conditions x ˆ (0> w) = x0 (w) >
(29)
³ ˆ (0) > ˆ (O) = eO (0) > (0) > ˆ (O) ˆ (0) > ´ ˆ (O) > (31) (O) > (O) > ˆ (O) >
Using (16), we obtain (21)
(28)
with boundary conditions (we omit the argument w for brevity) ³ ˆ (0) > ˆ (0) = e0 (0) > (0) > ˆ (0) + ˆ (0) > ´ ˆ (O) > (30) (O) > (O) > ˆ (O) >
¶
x + f + ( + ) @2) { = 0> w + (¯
(27)
(26)
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recalling the assumption that f À |¯ x|. The time derivative of Y along solutions of (32)—(34) is
2 1.8
L norm of observer error in u
1.6
˜ 2 (0) Y˙ = Y + ˜ 2 (0) 2
1.2 1 0.8
2
˜ (O) hO@(f¯x) = (37) ˜ 2 (O) hO@(f+¯x) +
1.4
At this point we need to select e0 and eO such ˜ 2 (0) and ˜ that ˜ 2 (0) 2 (O) hO@(f+¯x) + 2 ˜ (O) hO@(f¯x) are negative. We adopt the par ticularly simple choice made in Coron et al. (2004), and select e0 and eO such that (38)
˜ (O) = nO ˜ (O) >
(39)
0.4 0.2 0 0
5
10
15
20
25
30
20
25
30
Time (s) 2
x 10
6
1.8 1.6
L2 norm of observer error in p
˜ (0) > ˜ (0) = n0
0.6
1.4 1.2 1
0.8 0.6
in which case
0.4
¢ 2 ¡ ˜ (0) Y˙ = Y 1 n02 ³ ´ 2 O@(f¯ x) hO@(f+¯x) nO h ˜ 2 (O) = (40)
0.2 0 0
(41)
Y˙ Y=
(42)
10
15
Time (s)
Fig. 1. Observer error with (solid) and without (dashed) output injection.
So, if |n0 | 1 and |nO | hO@f
5
4. ADAPTATION OF FRICTION COEFFICIENT
then Adding friction to the model (4), we have the mass balance Cs Cx Cs +x + (n + s) = 0> (47) Cw C{ C{ and momentum conservation Cx f2 Cs i |x| x Cx +x + = (1 + ) > (48) Cw C{ n + s C{ 2 G where G is the pipe diameter, and is considered an unknown constant that accounts for uncertainty in the friction coe!cient i , which is given by Schetz and Fuhs (1996) "µ # ¶1=11 @G 6=9 1 s = 1=8log10 + = (49) 3=7 Reg i
Since (36) defines a norm equivalent to the O2 norm on [0> O], it follows that system (32) with (38)—(39) is exponentially stable at the origin in the O2 norm. Notice that (41) implies that whenever |nO | ? 1, there exists A 0 for which (42) holds. Replacing (26)—(27) with the new boundary conditions, the observer becomes C sˆ Cx ˆ C sˆ +x ˆ + (n + sˆ) = 0> Cw C{ C{ f C sˆ Cx ˆ Cx ˆ + +x ˆ = 0> Cw n + sˆ C{ C{
(43) (44)
@G is the pipe relative roughness, Reg is the Reynolds number defined as xG > (50) Reg = and is the fluid viscosity. The observer is then C sˆ Cx ˆ C sˆ +x ˆ + (n + sˆ) = 0> (51) Cw C{ C{ ³ ´ ˆ x| x Cx ˆ f2 C sˆ Cx ˆ ˆ ˆ i |ˆ +x ˆ + = 1+ > (52) Cw C{ n + sˆ C{ 2 G ˆ of > and which incorporates an estimate with boundary conditions (45)—(46). Consider the heuristic parameter update law ³ ´ ˜ (0) > ˆ˙ = ˜ (O) + (53)
with boundary conditions x ˆ (0) = x (0) + f
1 n0 ln 1 + n0
µ
n + s (0) n + sˆ (0)
¶ >
(45)
sˆ (O) = (n + s (O)) µ ¶ nO 1 × exp (x (O) x ˆ (O)) n= (46) f (1 + nO ) When n0 = 1 and nO = 1, (45)—(46) reduces to (26)—(27). It is interesting to notice that the above Lyapunov analysis does not provide exponential convergence in this case. Another interesting observation to make is that the design is independent of the working condition (¯ x> s¯). Figure 1 shows the observer error in terms of evolution in time of the ˆ ({> w) for the cases O2 (0> O) norm of x ({> w) x with and without output injection (the O2 norm of s ({> w) sˆ ({> w) looks qualitatively the same).
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where is a strictly positive constant. In physical coordinates, equation (53) corresponds to ˆ˙ = (*1 + *2 ) > (54)
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ADCHEM 2006
6
1
5
0.8
4
Leak size (kg/s)
Error in estimate of '
1.2
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-1
5
Time (min)
1
1.5
2
2.5
3
2500
¶ n + sˆ (0) > (55) ˆ (0) + f ln *1 = x (0) x n + s (0) ¶ µ n + s (O) *2 = x (O) x = (56) ˆ (O) + f ln n + sˆ (O) µ
2400 2300
Leak position (m)
2200 2100 2000 1900
ˆ when the Figure 2 shows the evolution of initial friction in the observer is twice that of the plant.
1800 1700 1600
0.5
1
1.5
2
2.5
3
Fig. 4. Estimated position of leak (solid) and actual position of leak (dashed). and z > { and are strictly positive constants. In physical coordinates, equation (62)—(63) corresponds to
Adding a leak to the model (47)—(48), with = 0> we have the mass balance Cs Cx f2 Cs +x + (n + s) = io ({) > (57) Cw C{ C{ D and the momentum conservation Cx f2 Cs i |x| x 1 f2 Cx +x + = + xio ({) > Cw C{ n + s C{ 2 G Dn+s (58) where D is the pipe cross sectional area. Assuming a point leak, we select io ({) as io ({) = zo ({ {o ) >
0
Time (min)
5. LEAK DETECTION
z ˆ˙ o = z (*1 *2 ) > 1 1 { ˆ˙ o = { (*1 + *2 ) |*1 + *2 | >
(59)
The leak detection test in the previous section was a nominal test, where the plant dynamics and the observer dynamics were identical (except for output injection, of course). Here, we perform more realistic tests, replacing the plant dynamics by the state-of-the-art flow simulator OLGA2000 1 . For two di erent cases, summarized in Table 1, we run our leak detection scheme (60)—(63) with (45)— (46). In the table, zlq denotes the mass rate of fluid at the inlet. Figure 5 shows that the leaks are quantified very accurately, while localization is somewhat noisy. However, the average error in position taken over the last 15 seconds shown in the Figure, is within 0=25% and 0=36% of the pipe length for Cases I and II, respectively.
which incorporates estimates of the leak size and ˆo . Consider the heuristic parameter position, z ˆo , { update laws ³ ´ ˜ (0) ˜ (O) > (62) z ˆ˙ o = z (63)
˜ (0) > ˜ (O) + * =
(64)
(66)
6. SIMULATIONS WITH OLGA
Cx ˆ f2 C sˆ Cx ˆ +x ˆ + Cw C{ n + sˆ C{ 1 f2 x| x ˆ iˆ |ˆ + x ˆz ˆo ({ { ˆo ) > (61) = 2 G D n + sˆ
¯ ¯ 1 1 { ˆ˙ o = { * ¯* ¯ >
(65)
where *1 and *2 are given in (55)—(56). Figures 3—4 show the evolution of the estimates (65)—(66) for a leak occuring at w = 0=25 minutes.
where zo and {o are the size of the leak and position of the leak, respectively, and denotes the Dirac distribution. The observer is then C sˆ Cx ˆ f2 C sˆ +x ˆ + (n + sˆ) = z ˆo ({ { ˆo ) > (60) Cw C{ C{ D
IFAC
0.5
Fig. 3. Estimated size of leak (solid) and actual size of leak (dashed).
where
where
0
Time (min)
Fig. 2. Error in estimated friction factor, that is ˆ .
and
2
1
0
-0.2
3
1
OLGA2000 is a commercially available flow simulator widely used by the petroleum industry. It is developed by Scandpower AS.
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Parameter O G f n ˆ { zo {o zlq
Case I 990 0=10 1=26h3 1=38h9 0=0063 0 0=13 4=9 505 70
Case II 8000 0=51 1=2h3 1=24h9 0=0056 0 0=028 12=5 4000 300
REFERENCES
Unit m m m/s Pa Pa s m kg/s m kg/s
L. Billmann and R. Isermann, "Leak detection methods for pipelines," Automatica, vol. 23, no. 3, pp. 381—385, 1987. J.-M. Coron, B. d’Andrea-Novel, and G. Bastin, "A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws," Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, December 14-17, 2004. EFA Technologies, Inc., "Events at a leak," Technical paper available from EFA on request, 1987. EFA Technologies, Inc., "E ect of pressure measurement resolution on PPA leak detection," Technical paper available from EFA on request, 1990. EFA Technologies, Inc., "PPA event suppression capability," Technical paper available from EFA on request, 1991. Fantoft, "D-SPICE overview: Production management systems," Fantoft Process Technologies, 2005. J.-M. Greenberg and T.-t. Li, "The e ect of boundary damping for the quasilinear wave equations," Journal of Di erential Equations, vol. 52, pp. 66-75, 1984. J. de Halleux, C. Prieur, J.-M. Coron, B. d’Andrea-Novel, and G. Bastin, "Boundary feedback control in networks of open channels," Automatica, vol. 39, pp. 1365—1376, 2003. M. Hou and P. Müller, "Fault detection and isolation observers," International Journal of Control, vol. 60, no. 5, pp. 827—846, 1994. A.O. Nieckele, A.M.B. Brage, and L.F.A. Azevedo, "Transient pig motion through gas and liquid pipelines," Journal of Energy Resources Technology, vol. 123, no. 4, 2001. J. Salvesen, Leak Detection by Estimation in an Oil Pipeline, MSc Thesis, NTNU, 2005. J.A. Schetz and A.E. Fuhs, Handbook of Fluid Dynamics and Fluid Machinery, Volume 1, Fundamentals of Fluid Mechanics, John Wiley & Sons, 1996. D.N. Shields, S.A. Ashton, and S. Daley, "Design of nonlinear observers for detecting faults in hydraulic sub-sea pipelines," Control Engineering Practice, vol. 9, pp. 297—311, 2001. C. Verde, "Multi-leak detection and isolation in fluid pipelines," Control Engineering Practice, vol. 9, pp. 673—682, 2001. C. Verde, "Minimal order nonlinear observer for leak detection," Journal of Dynamic Systems, Measurement and Control, vol. 126, pp. 467— 472, 2004. C.-Z. Xu and G. Sallet, "Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems," ESAIM Journal on Control, Optimisation and Calculus of Variations, vol. 7, pp. 421-442, 2002.
Table 1. Numerical coe!cients. Case I 6
Leak size (kg/s)
5 4 3 2 1 0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.3
0.35
0.4
0.45
0.5
Time (min)
Leak position (m)
550 500 450 400 350 300
0
0.05
0.1
0.15
0.2
0.25
Time (min)
Case II Leak size (kg/s)
20 15 10 5 0 0
0.5
1
1.5
2
2.5
3
3.5
4
2.5
3
3.5
4
Time (min)
Leak position (m)
4500 4000 3500 3000 2500
0
0.5
1
1.5
2
Time (min)
Fig. 5. Leak detection applied to OLGA simulations. 7. CONCLUDING REMARKS We have designed a leak detection system for pipelines consisting of an adaptive Luenbergertype observer and heuristic update laws for the parameters characterizing a point leak. The only available process information is flow velocity and pressure at the inlet and outlet of the pipe. Simulations with a state-of-the-art flow simulator as process, demonstrate accurate quantification and localization in two test cases. ACKNOWLEDGEMENTS We gratefully acknowledge the support from the Gas Technology Center at NTNU, Statoil, and the Norwegian Research Council.
IFAC
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ADCHEM 2006
OBSERVER DESIGN USING BOUNDARY INJECTIONS FOR PIPELINE MONITORING AND LEAK DETECTION Ole Morten Aamo, Jørgen Salvesen, and Bjarne A. Foss
Department of Engineering Cybernetics Norwegian University of Science and Technology N-7491 Trondheim
Abstract: We design a leak detection system consisting of an adaptive Luenbergertype observer based on a set of two coupled one dimensional first order nonlinear hyperbolic partial di erential equations governing the flow dynamics. It is assumed that measurements are only available at the inlet and outlet of the pipe, and output injection is applied in the form of boundary conditions. For the linearized model without friction and leak, exponential convergence of the state estimates to the plant state is shown by Lyapunovs method. The observer design is performed for the continuum model, ensuring that convergence properties established theoretically are not an artifact of the method of discretization. Heuristic update laws for adaptation of the friction coe!cient and leak parameters are given, and simulations demonstrate their ability to detect, quantify and locate leaks. Keywords: Partial di erential equations; Observers; Adaptive systems; Pipeline leaks
1. INTRODUCTION
box), while others incorporate models based on physical principles. Our method falls into the latter category, in that we will use a dynamic model of the pipe flow based on a set of two coupled hyperbolic partial di erential equations.
Transportation of liquids in pipelines requires monitoring to detect malfunctioning such as leaks. In the petroleum industry, leaks from pipelines may potentially cause environmental damage, as well as economic loss. These are motivating factors, along with requirements from environmental authorities, for developing e!cient leak detection systems. While some leak detection methods are hardware-based, relying on physical equipment being installed along the pipeline, the focus of this paper is on software-based methods that work for cases with limited instrumentation. In fact, instrumentation in the petroleum industry is usually limited to the inlet and outlet of pipelines, only. This calls for sophisticated signal processing methods to obtain reliable detection of leaks. Some software-based leak detection methods perform statistical analysis on measurements (black
IFAC
There have been numerous studies on model based leak detection. We mention here the most relevant ones with regard to our work. Based on a discretized pipe flow model, Billman and Isermann (1987) designed an observer with friction adaptation. In the event of a leak, the outputs from the observer di ers from the measurements, and this is exploited in a correlation technique that detects, quantifies and locates the leak. Verde (2001) used a bank of observers, computed by the method for fault detection and isolation developed by Hou and Müller (1994). The underlying model is a linearized, discretized pipe flow model on a grid of Q nodes. The observers are designed in
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ADCHEM 2006
such a way that all but one will react to a leak. Which one of the Q observers that does not react to the leak depends on the position of the leak, and this is the mechanism by which the leak is located. The outputs of the remaining observers are used for quantifying the leak. The bank of observers are computed using the recursive numerical procedure suggested by Hou and Müller (1994), however it was shown in Salvesen (2005) that due to the simple structure of the discretized model, the observers may be written explicitly. This is important, because it removes the need for recomputing the bank of observers when the operating point of the pipeline is changed. Verde (2004) also proposed a nonlinear version, using an extremely coarse discretization grid.
and the momentum conservation (ignoring friction for now) Cx 1 Cs Cx +x + = 0> Cw C{ C{
(2)
for ({> w) 5 (0> O) × (0> 4), and where x is flow velocity, s is pressure, and is density. The relation between pressure and density is modelled as (Nieckele et al. (2001)) s suhi > (3) f2 where uhi is a reference density at reference pressure suhi , and f is the speed of sound. Equation (1)—(2) also describes gas flow in a pipe, simply by replacing (3) with the ideal gas law. Under the conditions we consider, we assume f is su!ciently large to ensure A 0= In compact form, we have ¸ ¸ C s C s + D (s> x) = 0> (4) Cw x C{ x = uhi +
Several companies o er commercial solutions to pipeline monitoring with leak detection. Fantoft (2005) uses a transient model approach in conjunction with the commercial pipeline simulator OLGA2000, while EFA Technologies (1987, 1990, 1991) uses an event detection method that looks for signatures of no-leak to leak transitions in the measurements.
where, using (3), 5
6 x n+s 8> D (s> x) = 7 f2 x n+s n = f2 uhi suhi >
The detection method of Verde (2001) using a bank of observers, can potentially detect multiple leaks. However, multiple simultaneous leaks is an unlikely event, so the complex structure of a bank of Q observers seems unnecessary. In this paper, we instead employ ideas from adaptive control, treating the size and location of a single point leak as constant unknown parameters. This differs from the method of Billman and Isermann (1987), since we will model the leak in the observer, thereby obtaining state estimates also in a leak situation. Another important aspect of our method is that the observer is designed for the continuum model, ensuring that convergence properties established theoretically are not an artifact of the method of discretization. Our leak detection system consists of an adaptive Luenbergertype observer, based on a set of two coupled one dimensional first order nonlinear hyperbolic partial di erential equations governing the flow dynamics. It is assumed that measurements are only available at the inlet and outlet of the pipe, and output injection is applied in the form of boundary conditions. Heuristic update laws for adaptation of the friction coe!cient and the two leak parameters are suggested.
(5) (6)
and the boundary conditions are x (0> w) = x0 (w) >
(7)
s (O> w) = sO (w) =
(8)
The eigenvalues of D are 1 = x f> 2 = x + f=
(9)
Assuming that x ¿ f> which is always the case in the applications we are considering, the eigenvalues are distinct and satisfy 1 ? 0 ? 2 =
(10)
The system is therefore strictly hyperbolic. Steady state solutions (¯ s> x ¯) of (4), must satisfy ¸ C s = 0= (11) D (s> x) C{ x Since D (s> x) is invertible (l 6= 0), we have that Cx ¯ C s¯ = 0> = 0> (12) C{ C{ so s¯ and x ¯ are constant. The boundary conditions ¯ = x0 . (7)—(8) yield s¯ = sO and x
2. MATHEMATICAL MODEL 2.2 Model in Characteristic Form 2.1 Physical Model Consider now the change of coordinates
For liquid flow in a pipe we have the mass conservation Cs Cx Cs +x + f2 = 0> (1) Cw C{ C{
IFAC
µ (s> x) = f ln
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¶ n+s +xx ¯> n + s¯
(13)
ADCHEM 2006
µ (s> x) = f ln
¶ n+s +xx ¯> n + s¯
sˆ (O> w) = sO (w) =
(14)
Notice that the input to (4)—(8) is also copied in (25)—(27). Equation (25)—(27) can be viewed as a Luenberger-type observer, and convergence is guaranteed when the process is operated at asymptotically stable fixed points, which is usually the case for pipelines. However, we look for alternatives to the boundary conditions (26)—(27) which yield better convergence properties. Taking s0 (w) = s (0> w) and xO (w) = x (O> w) as process measurements, we may apply output injection to (25)—(27). In transformed coordinates, we obtain
which clearly is defined for all physically feasible s and x. It is easy to see that it’s inverse is s (> ) = (n + s¯) exp (( ) @ (2f)) n> (15) x (> ) = x ¯ + ( + ) @2=
(16)
Notice that the fixed point (¯ s> x ¯) corresponds to (0> 0) in the new coordinates. The time derivative of (13)—(14) is sw + xw > n+s sw + xw = w = f n+s
w = f
³ ³ ´ ´ ˆ @2 ¯+f+ ˆ + ˆ { = 0> ˆw + x ³ ³ ´ ´ ˆ @2 ˆ = 0> ˆ + x ¯f+ ˆ + w {
(17) (18)
Inserting for sw and xw from (4) yields µ
xs{ + (n + s) x{ f2 s{ + xx{ n+s n+s ¶ µ s{ + x{ = (x + f) f n+s = (x + f) { > (19) µ 2 ¶ xs{ + (n + s) x{ f s{ + xx{ w = f n+s n+s ¶ µ s{ + x{ = (x f) f n+s = (x f) { = (20)
w = f
x f + ( + ) @2) { = 0= w + (¯
(22)
where e0 and eO are functions to be designed. Notice that the boundary injections (30)—(31) may be any function of the known signals at both ends of the pipe. For convergence analysis, we consider the linearization of (21)—(22) and (28)— (29) around (0> 0) and form the dynamics of the ˜ = = ˆ observer error, defined as ˜ = ˆ, We obtain ¸ ¸ C C ˜ ˜ (32) ˜ + C{ ˜ = 0> Cw
The boundary conditions are obtained from (15)— (16), and are (0> w) + (0> w) = 0>
(23)
(O> w) (O> w) = 0=
(24)
with boundary conditions ³ ˆ (0) > ˜ (0) = e0 (0) > (0) > ˆ (0) > ˜ (0) + ´ ˆ (O) > (33) (O) > (O) > ˆ (O) >
The characteristic form (21)—(22) is convenient for the observer design carried out in the next section.
³ ˆ (0) > ˜ (O) = eO (0) > (0) > ˜ (O) ˆ (0) > ´ ˆ (O) > (34) (O) > (O) > ˆ (O) >
3. OBSERVER DESIGN In reality, input signals to pipelines are usually choke openings at the inlet and outlet. Here, we instead view x0 (w) and sO (w) in (7)—(8) as inputs to the process, and construct the copy of the plant dynamics (4) ¸ ¸ C sˆ C sˆ + D (ˆ s> x ˆ) = 0> (25) ˆ ˆ Cw x C{ x
where
IFAC
=
¸ x ¯+f 0 = 0 x ¯f
(35)
Following Xu and Sallet (2002) and Coron et al. (2004), consider the Lyapunov function candidate Z O 1 ˜ 2 h{@(f+¯x) g{ Y = f+x ¯ 0 Z O 1 ˜ 2 h{@(f¯x) g{> (36) + fx ¯ 0
with boundary conditions x ˆ (0> w) = x0 (w) >
(29)
³ ˆ (0) > ˆ (O) = eO (0) > (0) > ˆ (O) ˆ (0) > ´ ˆ (O) > (31) (O) > (O) > ˆ (O) >
Using (16), we obtain (21)
(28)
with boundary conditions (we omit the argument w for brevity) ³ ˆ (0) > ˆ (0) = e0 (0) > (0) > ˆ (0) + ˆ (0) > ´ ˆ (O) > (30) (O) > (O) > ˆ (O) >
¶
x + f + ( + ) @2) { = 0> w + (¯
(27)
(26)
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recalling the assumption that f À |¯ x|. The time derivative of Y along solutions of (32)—(34) is
2 1.8
L norm of observer error in u
1.6
˜ 2 (0) Y˙ = Y + ˜ 2 (0) 2
1.2 1 0.8
2
˜ (O) hO@(f¯x) = (37) ˜ 2 (O) hO@(f+¯x) +
1.4
At this point we need to select e0 and eO such ˜ 2 (0) and ˜ that ˜ 2 (0) 2 (O) hO@(f+¯x) + 2 ˜ (O) hO@(f¯x) are negative. We adopt the par ticularly simple choice made in Coron et al. (2004), and select e0 and eO such that (38)
˜ (O) = nO ˜ (O) >
(39)
0.4 0.2 0 0
5
10
15
20
25
30
20
25
30
Time (s) 2
x 10
6
1.8 1.6
L2 norm of observer error in p
˜ (0) > ˜ (0) = n0
0.6
1.4 1.2 1
0.8 0.6
in which case
0.4
¢ 2 ¡ ˜ (0) Y˙ = Y 1 n02 ³ ´ 2 O@(f¯ x) hO@(f+¯x) nO h ˜ 2 (O) = (40)
0.2 0 0
(41)
Y˙ Y=
(42)
10
15
Time (s)
Fig. 1. Observer error with (solid) and without (dashed) output injection.
So, if |n0 | 1 and |nO | hO@f
5
4. ADAPTATION OF FRICTION COEFFICIENT
then Adding friction to the model (4), we have the mass balance Cs Cx Cs +x + (n + s) = 0> (47) Cw C{ C{ and momentum conservation Cx f2 Cs i |x| x Cx +x + = (1 + ) > (48) Cw C{ n + s C{ 2 G where G is the pipe diameter, and is considered an unknown constant that accounts for uncertainty in the friction coe!cient i , which is given by Schetz and Fuhs (1996) "µ # ¶1=11 @G 6=9 1 s = 1=8log10 + = (49) 3=7 Reg i
Since (36) defines a norm equivalent to the O2 norm on [0> O], it follows that system (32) with (38)—(39) is exponentially stable at the origin in the O2 norm. Notice that (41) implies that whenever |nO | ? 1, there exists A 0 for which (42) holds. Replacing (26)—(27) with the new boundary conditions, the observer becomes C sˆ Cx ˆ C sˆ +x ˆ + (n + sˆ) = 0> Cw C{ C{ f C sˆ Cx ˆ Cx ˆ + +x ˆ = 0> Cw n + sˆ C{ C{
(43) (44)
@G is the pipe relative roughness, Reg is the Reynolds number defined as xG > (50) Reg = and is the fluid viscosity. The observer is then C sˆ Cx ˆ C sˆ +x ˆ + (n + sˆ) = 0> (51) Cw C{ C{ ³ ´ ˆ x| x Cx ˆ f2 C sˆ Cx ˆ ˆ ˆ i |ˆ +x ˆ + = 1+ > (52) Cw C{ n + sˆ C{ 2 G ˆ of > and which incorporates an estimate with boundary conditions (45)—(46). Consider the heuristic parameter update law ³ ´ ˜ (0) > ˆ˙ = ˜ (O) + (53)
with boundary conditions x ˆ (0) = x (0) + f
1 n0 ln 1 + n0
µ
n + s (0) n + sˆ (0)
¶ >
(45)
sˆ (O) = (n + s (O)) µ ¶ nO 1 × exp (x (O) x ˆ (O)) n= (46) f (1 + nO ) When n0 = 1 and nO = 1, (45)—(46) reduces to (26)—(27). It is interesting to notice that the above Lyapunov analysis does not provide exponential convergence in this case. Another interesting observation to make is that the design is independent of the working condition (¯ x> s¯). Figure 1 shows the observer error in terms of evolution in time of the ˆ ({> w) for the cases O2 (0> O) norm of x ({> w) x with and without output injection (the O2 norm of s ({> w) sˆ ({> w) looks qualitatively the same).
IFAC
where is a strictly positive constant. In physical coordinates, equation (53) corresponds to ˆ˙ = (*1 + *2 ) > (54)
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6
1
5
0.8
4
Leak size (kg/s)
Error in estimate of '
1.2
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
-1
5
Time (min)
1
1.5
2
2.5
3
2500
¶ n + sˆ (0) > (55) ˆ (0) + f ln *1 = x (0) x n + s (0) ¶ µ n + s (O) *2 = x (O) x = (56) ˆ (O) + f ln n + sˆ (O) µ
2400 2300
Leak position (m)
2200 2100 2000 1900
ˆ when the Figure 2 shows the evolution of initial friction in the observer is twice that of the plant.
1800 1700 1600
0.5
1
1.5
2
2.5
3
Fig. 4. Estimated position of leak (solid) and actual position of leak (dashed). and z > { and are strictly positive constants. In physical coordinates, equation (62)—(63) corresponds to
Adding a leak to the model (47)—(48), with = 0> we have the mass balance Cs Cx f2 Cs +x + (n + s) = io ({) > (57) Cw C{ C{ D and the momentum conservation Cx f2 Cs i |x| x 1 f2 Cx +x + = + xio ({) > Cw C{ n + s C{ 2 G Dn+s (58) where D is the pipe cross sectional area. Assuming a point leak, we select io ({) as io ({) = zo ({ {o ) >
0
Time (min)
5. LEAK DETECTION
z ˆ˙ o = z (*1 *2 ) > 1 1 { ˆ˙ o = { (*1 + *2 ) |*1 + *2 | >
(59)
The leak detection test in the previous section was a nominal test, where the plant dynamics and the observer dynamics were identical (except for output injection, of course). Here, we perform more realistic tests, replacing the plant dynamics by the state-of-the-art flow simulator OLGA2000 1 . For two di erent cases, summarized in Table 1, we run our leak detection scheme (60)—(63) with (45)— (46). In the table, zlq denotes the mass rate of fluid at the inlet. Figure 5 shows that the leaks are quantified very accurately, while localization is somewhat noisy. However, the average error in position taken over the last 15 seconds shown in the Figure, is within 0=25% and 0=36% of the pipe length for Cases I and II, respectively.
which incorporates estimates of the leak size and ˆo . Consider the heuristic parameter position, z ˆo , { update laws ³ ´ ˜ (0) ˜ (O) > (62) z ˆ˙ o = z (63)
˜ (0) > ˜ (O) + * =
(64)
(66)
6. SIMULATIONS WITH OLGA
Cx ˆ f2 C sˆ Cx ˆ +x ˆ + Cw C{ n + sˆ C{ 1 f2 x| x ˆ iˆ |ˆ + x ˆz ˆo ({ { ˆo ) > (61) = 2 G D n + sˆ
¯ ¯ 1 1 { ˆ˙ o = { * ¯* ¯ >
(65)
where *1 and *2 are given in (55)—(56). Figures 3—4 show the evolution of the estimates (65)—(66) for a leak occuring at w = 0=25 minutes.
where zo and {o are the size of the leak and position of the leak, respectively, and denotes the Dirac distribution. The observer is then C sˆ Cx ˆ f2 C sˆ +x ˆ + (n + sˆ) = z ˆo ({ { ˆo ) > (60) Cw C{ C{ D
IFAC
0.5
Fig. 3. Estimated size of leak (solid) and actual size of leak (dashed).
where
where
0
Time (min)
Fig. 2. Error in estimated friction factor, that is ˆ .
and
2
1
0
-0.2
3
1
OLGA2000 is a commercially available flow simulator widely used by the petroleum industry. It is developed by Scandpower AS.
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Parameter O G f n ˆ { zo {o zlq
Case I 990 0=10 1=26h3 1=38h9 0=0063 0 0=13 4=9 505 70
Case II 8000 0=51 1=2h3 1=24h9 0=0056 0 0=028 12=5 4000 300
REFERENCES
Unit m m m/s Pa Pa s m kg/s m kg/s
L. Billmann and R. Isermann, "Leak detection methods for pipelines," Automatica, vol. 23, no. 3, pp. 381—385, 1987. J.-M. Coron, B. d’Andrea-Novel, and G. Bastin, "A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws," Proceedings of the 43rd IEEE Conference on Decision and Control, Paradise Island, Bahamas, December 14-17, 2004. EFA Technologies, Inc., "Events at a leak," Technical paper available from EFA on request, 1987. EFA Technologies, Inc., "E ect of pressure measurement resolution on PPA leak detection," Technical paper available from EFA on request, 1990. EFA Technologies, Inc., "PPA event suppression capability," Technical paper available from EFA on request, 1991. Fantoft, "D-SPICE overview: Production management systems," Fantoft Process Technologies, 2005. J.-M. Greenberg and T.-t. Li, "The e ect of boundary damping for the quasilinear wave equations," Journal of Di erential Equations, vol. 52, pp. 66-75, 1984. J. de Halleux, C. Prieur, J.-M. Coron, B. d’Andrea-Novel, and G. Bastin, "Boundary feedback control in networks of open channels," Automatica, vol. 39, pp. 1365—1376, 2003. M. Hou and P. Müller, "Fault detection and isolation observers," International Journal of Control, vol. 60, no. 5, pp. 827—846, 1994. A.O. Nieckele, A.M.B. Brage, and L.F.A. Azevedo, "Transient pig motion through gas and liquid pipelines," Journal of Energy Resources Technology, vol. 123, no. 4, 2001. J. Salvesen, Leak Detection by Estimation in an Oil Pipeline, MSc Thesis, NTNU, 2005. J.A. Schetz and A.E. Fuhs, Handbook of Fluid Dynamics and Fluid Machinery, Volume 1, Fundamentals of Fluid Mechanics, John Wiley & Sons, 1996. D.N. Shields, S.A. Ashton, and S. Daley, "Design of nonlinear observers for detecting faults in hydraulic sub-sea pipelines," Control Engineering Practice, vol. 9, pp. 297—311, 2001. C. Verde, "Multi-leak detection and isolation in fluid pipelines," Control Engineering Practice, vol. 9, pp. 673—682, 2001. C. Verde, "Minimal order nonlinear observer for leak detection," Journal of Dynamic Systems, Measurement and Control, vol. 126, pp. 467— 472, 2004. C.-Z. Xu and G. Sallet, "Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems," ESAIM Journal on Control, Optimisation and Calculus of Variations, vol. 7, pp. 421-442, 2002.
Table 1. Numerical coe!cients. Case I 6
Leak size (kg/s)
5 4 3 2 1 0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.3
0.35
0.4
0.45
0.5
Time (min)
Leak position (m)
550 500 450 400 350 300
0
0.05
0.1
0.15
0.2
0.25
Time (min)
Case II Leak size (kg/s)
20 15 10 5 0 0
0.5
1
1.5
2
2.5
3
3.5
4
2.5
3
3.5
4
Time (min)
Leak position (m)
4500 4000 3500 3000 2500
0
0.5
1
1.5
2
Time (min)
Fig. 5. Leak detection applied to OLGA simulations. 7. CONCLUDING REMARKS We have designed a leak detection system for pipelines consisting of an adaptive Luenbergertype observer and heuristic update laws for the parameters characterizing a point leak. The only available process information is flow velocity and pressure at the inlet and outlet of the pipe. Simulations with a state-of-the-art flow simulator as process, demonstrate accurate quantification and localization in two test cases. ACKNOWLEDGEMENTS We gratefully acknowledge the support from the Gas Technology Center at NTNU, Statoil, and the Norwegian Research Council.
IFAC
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