ARTICLES

Online implementation of a leak isolation algorithm in a plastic pipeline prototype

O. Begovich^†, A. Pizano-Moreno ^†, and G. Besançon^‡

^† CINVESTAV, Unidad Guadalajara, Guadalajara, Jalisco, Mexico, C.P. 45019
obegovi@gdl.cinvestav.mx
^‡ Gipsa-lab, Institut Universitaire de France, Grenoble INP BP 46 38402 Saint-Martin d'Heres, France.
gildas.besancon@gipsa-lab.grenoble-inp.fr

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Abstract - In this paper, a real-time application of a Leak Detection and Isolation (LDI) algorithm for a plastic pipeline is presented. This LDI algorithm is fed with flow and pressure signals coming from sensors placed at the ends of the pipeline. It uses a flow observer based on a model obtained from the Method of Characteristics, and it is designed in order to assure an acceptable real-time leak isolation by taking into account various practical difficulties. In particular it incorporates an adaptation law for the friction coefficient in order to compensate for possible variations. The whole scheme is successfully tested on a plastic pipeline prototype, transporting water and built as a possible benchmark.

Keywords - Fault Diagnosis; Leak Detection; Nonlinear Estimation; Plastic Pipelines; Water.

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I. INTRODUCTION

Nowadays, freshwater requires special attention in order to avoid its waste and pollution (UN/WWAP, 2003). Regarding the first aspect, implementation of pipeline monitoring systems to automate the detection and isolation of leaks is an important issue. Leak detection can obviously be easily realized by straight considerations, for instance as simple as a mass balance, i. e. just looking for any difference between the inlet and outlet flows but the real challenge is to find the exact position of a leak (leak isolation), as it will be seen along this paper.

As it is well known, a leak can produce damage to environment, water waste and economic effect. The methods most commonly used to locate leaks are the so-called Direct Methods. In those methods the leak is searched directly along the pipeline (Scott and Barrufet, 2003). This search can be just a visual inspection or by using some sophisticated equipment. Some direct leak detection methods are, for example, acoustic leak detection systems, infra-red thermography, ultrasonic methods or electromagnetic techniques as reported in Eiswirth and Burn (2001). The use of such systems allows to locate a leak in a precise way, but the price to pay for it is expensive equipment and a long time to locate the leak, meaning high costs, water waste and sometimes service disruption due to the necessity to empty the pipes for inspection.

An alternative way to detect and isolate leaks is by the analysis of the transient pressure waves. When a leak appears a pressure wave is generated, which travels throughout the pipeline and reflects from the boundaries as well as from the leak. This response can be analyzed to determine the location of the leak as in Misiunas et al. (2005) and Covas et al. (2005). A complete survey about these techniques can be found in Colombo et al. (2009).

Another commonly used LDI technique is the one known as Analytical or Model-based Method (API, 2007; Kowalczuc and Gunawickrama, 2004). In such a technique, a model of the system under study is necessary in order to detect and isolate a leak. Based in that model, an entity, such as an observer, a Kalman filter or an adaptive system, can be designed to detect any discrepancy between the predicted model behaviour and the measured one, in order to isolate the leak. Such an entity is usually implemented in a digital computer system as an algorithm whose inputs are available measurements from the monitored system and the outputs of this algorithm are both an alarm signal, when a leak is detected, and the leak parameters (location and magnitude). About the measurements, such LDI algorithms generally require flow and pressure signals coming from sensors placed at the ends of a pipeline, since they are available in many long pipeline systems, such as oil pipelines or aqueducts. In this work, a copy of the model in the leak-free case is directly used to design an observer (state estimator), whose input signals are the inlet and outlet pressure heads and the output signals are the estimated inlet and outlet flows. These estimates are then compared with the actual pipeline flow measurements, in order to generate the so-called residuals (i.e. the difference between the measured an the estimated flows). When the system operates normally the estimated and measured signals must be practically equal. When a leak appears a deviation between the estimated values and the measured ones appears. Thus such residuals are monitored to decide whether a leak has appeared. Once a leak is detected, the program triggers an alarm (leak detection) and the process to estimate the leak position is started (leak isolation). In general, such analytical LDI systems are very sensitive to modelling errors and noise, then observers must be designed using a good model and supplied with adequate information to achieve good position estimation. Otherwise, false alarms could be triggered and/or a bad leak isolation will be obtained. This makes the design of real-time implementable LDI algorithms still a challenge, and currently several research teams around the world are working on such a problem. The reader is encouraged to consult Chap. 22 of Kowalczuc and Gunawickrama (2004) where more details about these methods are discussed.

In the literature however, even if several analytical methods are used to locate pipeline leaks, a lot of them are implemented only in simulation, i.e. the LDI algorithms are fed with information from a software representing a hypothetical pipeline as in Kowalczuc and Gunawickrama (2004), Begovich et al. (2007). In other studies the results are obtained using online data coming from pilot pipelines as in Verde (2004), and Billmann and Isermann (1987) but without specifying if the LDI system was tested directly on the pipe (i.e. with a real online application), nor commenting about the specific difficulties found with such a real application. Moreover, the vast majority of the studied pipelines are concerned with highly rough pipes, such as metallic or concrete ones, where, in particular, a constant friction coefficient can be assured, which may not be the case in plastic pipelines.

From all this, a detailed study to assess the applicability of an LDI in a real pipeline system, where actual noises, uncertainty and neglected hydraulic phenomena are present, can be very useful for practitioners, and as far as we know there is no such study already available in a publication. On the other hand, in our opinion and experience, those analytical LDI methods provide a very good option for implementing a monitoring system over a real transportation area such as aqueducts (long pipeline systems carrying water from sources to cities).

The main contributions of this paper are: firstly, to improve a classical analytical LDI algorithm in order to be successfully implemented in an online application; secondly, to present details about an implementation of the designed LDI on a plastic pipeline prototype transporting water, with the aim to provide an appropriate reference for students and engineers working in this area. To do that, a full description of the used prototype is given and a detailed description of the LDI implementation is discussed. The difficulties found during the real-time implementation and their solutions are also commented.

The paper is organized as follows: In Section 2, a description of the used pipeline prototype and its characteristics are given. Some corresponding dynamical modelling is then stated in Section 3. In Section 4 the chosen LDI algorithm is briefly discussed and the proposed modifications are described. The LDI implementation on the prototype and the obtained results are shown in Section 5. Finally, conclusions and future work are presented in Section 6.

II. PIPELINE PROTOTYPE

With the purpose to implement an LDI monitoring system on a Mexican aqueduct in the near future and to evaluate LDI algorithms (as well as to develop new ones, for instance as in Torres et al. (2008)and Besancon et al. (2007)), a plastic pipeline prototype with short length and small diameter has been built at the Mexican Research and Advanced Studies Center (CIN-VESTAV), and is used in this work. This laboratory pipeline is an 80 m long rectangular closed circuit, as shown in Fig. 1. The pipes of PCR (Polypropylene Co-polymer Random) have an external diameter of 3" (0.0762m) and an internal diameter of 2.5" (0.0635m). This material has been chosen firstly, because of its low cost and weight, its easy handling, and its good weather resistance and secondly, because plastic pipes are more and more used, and consequently motivates specific LDI studies, facing its specific features.

Figure 1: Schematic diagram of the pipeline prototype.

For the experiments in this paper, all pipes are at the same level. However, thanks to the design of their supports, it is possible to change their inclinations. This will allow to test robustness of LDI systems faced to different slopes in other studies. A full description of the prototype goes as follows: A 750 1 container (E-l) is used as a water source as well as a final deposit. A 5 hp electrical pump (E-2) provides the energy needed to transport the water around the pipeline. The pump is controlled by a Variable-Frequency Drive (VFD) which controls the rotational speed of the pump motor by a variation on the AC frequency. The motor velocity changes are translated into inlet pressure changes. The instrumentation consists in flow sensors (Fl, F2) as well as pressure ones (PI, P2) installed at both ends of the pipeline. The flow rate is measured using electromagnetic flow meters (Endress Hauser™, mod. Promag 10W ) calibrated for an operation range of 0 to 6 1/s. The pressure head is measured using a piezometric diaphragm sensor (Endress Hauser™, mod. Cerabar M) calibrated for an operation range from 0 to 7 m. The four sensors deliver an output current of 4 to 20 mA. In the presented case, the pressure sensor has a rise time of 60 ms, and the flow sensor a measuring period of 50 ms. In order to ensure measuring accuracy, the sensors must be placed at least at 0.30 m away from a pump or pipeline elbow. In Fig. 1 the location of the sensors in this prototype is shown.

To emulate the effect of a leak, three 1/2" (0.0127m) Honeywell™ Servo valves mod. M7284A (V-l, V-2, V-3) were installed at 16.15 m, 32.45 m and 48.75 m (distances are relative with respect to the position of the first pair of sensors). These valves have an opening time of 30 s, and they are driven by a current of 4 to 20 mA. A computer in a control room is equipped with DAQ hardware (National Instruments™, mod. PCI-6014). This DAQ has 15 analogical voltage inputs and 2 analogical voltage outputs, and is used to sample the sensor signals and to control the opening of the servo valves installed in the system. Since the sensor outputs are current signals, a circuit to convert from current to voltage was built for each DAQ input. The servo valves remote control is achieved by using voltage to current converters. These converters are regulated by the analogical outputs of the DAQ. The LDI was implemented within Mathworks Simulink™ software. To guarantee real time measurements, Matlab's ™ tool Real Time Windows Target™ is used.

The prototype parameters are summarized in Table 1. The corresponding dynamical modelling is presented in the next section. Several pictures of the prototype can be found in Cinvestav (2010).

Table 1: Parameters of the pipeline prototype.

III. PIPELINE MODELING

Let us consider a one-dimensional isothermal model, where it is assumed that the fluid is slightly compressible, the walls of the pipeline are linearly elastic and are slightly deformable. If the variation of liquid density and the variation of internal area caused by a variation of the inside pressure are neglected, then the dynamics of the pipeline fluid can be described by the following partial differential equations (Roberson et al, 1998):

	(1)
	(2)

where R = ƒ /2DA_t, Q is the flow rate (m³ / s), H is the pressure head (m), z is the length coordinate (m), t is the time coordinate (s), g is the acceleration due to gravity (m / s²), A_t is the pipeline cross-section area (m²), D is the pipeline diameter (m), b is the speed of the pressure wave in the fluid (m / s) and ƒ is the friction coefficient. Notice that usually the sound speed in water, in metallic pipes, is used instead of the pressure wave speed b. But for our plastic pipeline, the actual pressure wave value is calculated as:

(3)

where e is the wall thickness, E is the modulus of elasticity of the pipeline material, ρ is the density of water and K is water's bulk modulus of elasticity, all these parameters are shown in Table 1.

However, because of its complexity, the model given by (l)-(2) cannot be directly used to design an analytic LDI System. To this aim, in the literature it is common to obtain lumped models from the discretization of (1) and (2). Those discretizations can be made in time and space by the use of methods such as Centered Finite Difference scheme (Kowalczuc and Gu-nawickrama, 2004; Billmann and Isermann, 1987), or the method of Characteristic Lines (Kowalczuc and Gunawickrama, 2004). These methods lead to a nonlinear discrete-time model which can be used to design, by example, an observer for LDI. In other works such as Verde (2004) and Besancon et al. (2007) equations (1) and (2) are discretized only in space resulting in an ordinary differential model. Finally, a simpler model can be obtained from (1) and (2) when the flow is assumed in steady state, i. e., when the head and flow variations with respect to time are considered close to zero. In this case, the model consists in algebraic equations as shown in Billmann and Isermann (1987).

Each type of discretization leads to a model having certain advantages and disadvantages: The Characteristic Lines discretization for instance is exact along the characteristic lines, subject to strictly satisfy the Courant's Condition (Roberson et al. 1998):

(4)

where Δz is the space step and Δt is the time step.

The Finite Difference scheme is more accurate if the discretization is finer and does not need to strictly satisfy Eq. (4). It is important that the selected scheme represents accurately the system behavior in steady state, since, this information is a key figure to achieve leak isolation. In this paper, the Characteristic Lines Method is chosen, basically because of its simplicity to solve partial differential equations, also because it results in low dimensional models meaning some easier industrial implementation.

Furthermore, model (l)-(2) is derived assuming a straight and uniform pipeline. However the considered prototype is a closed circuit and contains couplings and small metal sections where the servo valves and the sensors are installed. So, in order to achieve a better modeling, an equivalent straight pipe model is obtained thanks to a virtual substitution of each elbow, joint and tee by a segment of straight pipe presenting the same head loss as those parts. This transformation is important since if it is omitted, an incorrect or biased leak location could be estimated. The effective length of the prototype, i.e. the distance between sensors is 64.5 m and the equivalent straight length (Z) is 85 m. The equivalent straight position of the valves is shown in Table 2. Due to space limitations the transformation procedure is not recalled here, but the interested reader can consult Mott (2005) for details about it, and Garcia-Tirado et al. (2009) for an example where the pipeline model proposed in Verde (2004) is validated with data from our pipeline.

Table 2: Real and equivalent servo valve positions.

Finally, the value of the friction coefficient ƒ is related to the flow regime in the pipeline (see Mott, 2005): in the case of pipelines with high roughness (typically metal or concrete pipes), a completely turbulent flow is easily attained and consequently a constant value for ƒ is present. But in the case of plastic pipes, the roughness is closer to zero (see Willoughby, 2001) and then it is difficult to obtain a fully turbulent flow, which results in a value of ƒ sensitive to flow changes and thus makes an adaptive estimator for ƒ of interest. A very useful tool is the so-called Moody Diagram in which the sensitivity of ƒ can be determined in the presence of flow variations such as those generated by a change in the set point, noise, and leaks. In our setting a noncompletely turbulent flow is present, and ƒ is sensitive to flow changes as it will be shown in the following section.

IV. LDI ALGORITHM

Studies on real-time LDI algorithms viability are necessary since direct application of such algorithms require several adjustments. For example, in large aqueducts several phenomena might be present such as nonconstant water density, and friction and sinuous height profiles must be considered motivating both further studies on pilot pipelines and design of adjustable LDI algorithms such as the one presented here.

The algorithm tested in this work is based on the fault sensitive algorithm reported in Billmann and Is-ermann (1987), but with several modifications; first it was re-derived to use the model obtained from the Method of Characteristics. Secondly, it was enhanced with the online estimation of the friction coefficient by re-deriving the adaptation scheme of Kowalczuc and Gunawickrama (2004) for our modelling, in order to cope with variations of this coefficient. Notice that those variations can be explained by: a) The presence of a noncompletely turbulent flow regime due to the roughness close to zero in plastic pipes and the flow magnitude in our setting and b) The presence of set point changes and the occurrence of important leaks. Finally, the algorithm was improved by a friction coefficient freezing technique proposed here in order to avoid leak compensation by the friction adaptation. A scheme of the selected LDI method is depicted in Fig. 2. A general explanation of this LDI algorithm was given in introduction, and a more detailed description is given next, in terms of its main features:.

Figure 2: Scheme of the selected LDI algorithm.

Signal filtering. Unlike in simulation works, in many real applications the measured signals are corrupted with a considerable level of noise, which means that filters are needed together with an appropriate tuning of their parameters. The correct use of these filters allows detection of small leaks while if they are omitted, the LDI performance can become very poor. This in turn means, that in addition to the specific LDI parameters, that will be discussed in the sequel, the designer must face the tuning of all these filters according to the needs of his setting. Along this paper several general guidelines to adjust the synthesis parameters will be given so as to facilitate the designer task.

Nonlinear Estimator. In order to obtain the pipeline model used to design the nonlinear estimator, the discrete integration of (1) and (2) is carried out at the point (d,k) (see Fig. 3) using the Method of Characteristics as in Roberson et al. (1998) (note that in Kowalczuc and Gunawickrama (2004) and Billmann and Isermann (1987) the Finite Differences scheme is used). This discretization results from the integration along the positive and negative characteristic lines (shown in Fig. 3 and denoted by +b and -b respectively) which leads to a pair of equations in the form (5) and (6) for each node:

	(5)
	(6)

Figure 3: Discretization in time and space used by the Method of Characteristics.

where d = 0,1,..., iV, N = Z / Δz and t = k Δt and with boundary conditions:

(7)

These equations can be written as a model in state variable representation, as follows:

(8)

with

	(9)
	(10)

and where constant matrices A, B, D and E depend on the pipeline parameters and matrix C depends on the friction coefficient ƒ and some internal states. Details on those matrices for this particular application can be found in the Appendix. The output vector can be selected as:

(11)

Under the assumption that A^-1 exists, with equations (8) to (11) a simple nonlinear state estimator, to generate the residuals, is derived as:

	(12)
	(13)

This estimator can be transformed into an adaptive one by the inclusion of some online estimation of the friction factor ƒ, as this will be shown later on. Further improvements can be thought of, but in view of the good performances already obtained in practice with this scheme (and presented in next section), they are left for future developments.

Alarm (Leak Detection). In general, the direct use of the residuals deviation to activate the alarm is not adequate, since a false alarm can be triggered due to small deviations on the residuals generated by electrical and hydraulic noises. In order to avoid such false alarms, the detection is based on the cross-correlation of the flow rate residuals and where:

(14)

For a time shift , the cross-correlation computation is realized by a first order recursive filter:

(15)

where is a weighting factor between 0 and 1. In order to reduce noise effects, the alarm criterion is taken as the sum over several time shifts :

(16)

In the absence of any leak , but when a leak appears, the term becomes negative, (thus < 0). Therefore, the alarm is triggered when < a, where a is a negative threshold selected by the user. If the threshold is too small, the noise can activate the alarm (false alarm situation). Hence the smallest detectable leak depends on the noise present on the system.

Tuning Guideline 1: A good advice can be to select the threshold magnitude larger than the standard deviation of which can be computed from measurements of a steady state operation and without any leak. Obviously, this threshold value also depends on the characteristics of the pipeline in question. Notice from equation (16), that if the value of is increased then smaller leaks could be detected thanks to the accumulative effect of the sum, but in this case the leak detection time is increased. Notice also that in equation (15) the value of determines the sensibility of the alarm, in the sense that using a value near zero will allow a fast detection of a leak but the noise will be increased and a bigger threshold will be needed, while using a value near to one will allow to set a very small threshold but it will take a larger time to detect the leak. Finally notice that in this work, the correlation is calculated following the procedure shown in Billmann and Isermann (1987), but this correlation can be calculated using other algorithms such as the ones in Proakis and Manolakis (2006) as shown in Be-govich and Valdovinos-Villalobos (2010).

Leak Isolation. The leak isolation exploits the steady state pressure profile once the leak effect has stabilized (see Fig. 4). The leak location can be approximated as the crossing point of the pressure slopes using where values tan θ_in and tan θ_out are proportional to and respectively. Then, the location of a leak can be estimated by the following relationship where the square of the residuals have been calculated using auto-correlations:

(17)

Figure 4: Pressure profile of a pipeline in leak-free and leaking cases respectively.

where calculated as:

(18)

and

(19)

The corresponding value of μ is a constant between 0 and 1. Correlation is calculated in a similar way, but using and in the computation of . More details can be found in Kowalczuc and Gunawickrama (2004).

Tuning Guideline 2: For the tuning of (19) first note that if the parameter μ is close to 1 the resulting signal is very smooth since . The opposite happens if μ ≈ 0. In the former case the consequence is to have less noise sensitivity, but the cost to pay is a delay in the alarm activation. In short, this parameter allows to choose between noise immunity or a fast leak isolation time.

Tuning Guideline 3: The choice of a big value of results in less noise sensitivity but the cost to pay is again a delay in the triggering of the leak alarm. The opposite happens if is small.

The leak size can be estimated from a simple dynamic mass balance equation, with the use of an average as:

(20)

where the parameter is a time window used to calculate this average.

Friction Estimation. The first step is to determine the behavior of the friction coefficient, which can be done from the computation of the so-called Reynolds number (Roberson et al, 1998) as Re=V D/v, where V is the fluid velocity and v is the kinematic viscosity of the fluid. A typical flow rate in this pipeline is 0.0046 m³/s, v = 7 x 10^-6 for water at 20 C and Re = 92, 234. Using a Moody diagram it can be seen that the flow is not completely turbulent, meaning that the friction coefficient ƒ is not constant w.r.t. the flow rate, and thus must be estimated. In this work, the friction estimation is based on Kowalczuc and Gunawickrama (2004), and it is derived for the model used in this paper.

The estimation of ƒ is carried out when the pipe is in leak-free state in order to avoid leak effect compensation in the LDI. Here is assumed that the net effect of the fluid resistance is modeled by the single time varying parameter ƒ^k. The friction estimate is obtained as follows: Using (6) for node d = 0 and (5) for node d = N, two independent equations for the flow rates at the inlet and outlet can be derived:

	(21)
	(22)

these equations are linear in the parameter ƒ which allows to easily estimate it. Thus, two error functions can be formulated using the above quantities and their corresponding measurements.

	(23)
	(24)

minimizing the error functions with respect to ƒ

(25)

leads to two independent estimators for the friction coefficient.

	(26)
	(27)

The first one estimates the friction coefficient as seen from the inlet of the pipe, and the second one estimates the friction coefficient as seen from the outlet side. So, the two estimates can be combined into a unique value by using a first order recursive filter:

(28)

Equation (28) represents an iterative average for the friction coefficient and ζ is a weighting factor between 0 and 1.

Tuning Guideline 4 The selection of ζ ≈ 1 results in a very smooth value of and for ζ ≈ 0 the opposite happens. If the sensitivity of ƒ to flow changes is significant, as in our case, ζ close to 0 is recommended.

Friction coefficient freezing. Once a leak is detected the friction estimation must be frozen in order to avoid compensating for the leak. In our system, it is observed that, when the estimated friction is frozen immediately after the alarm is triggered, bad leak isolation is attained. A suitable way to freeze the friction estimate is obtained by the following average:

(29)

where t_a is the time when the leak has been detected and W, ω₀ are parameters of a time window. This window must take into account: a) the estimated value of ƒ before the leak starts, and b) the small deviation of the friction coefficient once the leak has started in order to attain an acceptable leak isolation.

Tuning Guideline 5: The parameters W and ω₀ must be determined experimentally, since they depend mainly on the friction sensitivity.

Notice that the pipeline model used for estimator (12)-(13) being stable, the estimation scheme is itself (at least locally) stable. This can formally be checked from a direct first order linearization analysis for instance. The addition of the proposed adaptation law can a priori affect this stability, but again it will be locally preserved, as it can be seen from a similar first order approximation analysis. In particular, this has been checked by analyzing the linearized model around several operating points for maximum and minimum values of the friction. In practice, the friction can be ini ialized from data sheets or calculated by the Swamee-Jain model (Swamee and Jain, 1976) in order to guarantee an appropriate initial value and a good convergence speed. The corresponding good behavior of the whole proposed scheme is actually confirmed by various simulation studies as well as real-time experiments, as it will be seen in next section.

V. EXPERIMENTAL RESULTS

The experiment presented in this section uses both flow and head measurements from sensors placed at the ends of the prototype and the LDI algorithm described in the previous section.

The first problem is the selection of the value of the parameters Δt and Δz. Since on the one hand, the time step Δt is restricted by the speed of the slowest sensor, in this case the pressure sensors, and on the other hand, the distance step Δz = Z/N must be chosen to satisfy Courant's Condition. Using (4) we have:

(30)

From (30), N must be chosen such that At is larger than the rise time of the pressure sensor. Selection of N = 2 leads to the suitable value Δt = 0.118 s (this value will be used as sampling time) and also to a low-dimensional model, as it can be seen in the Appendix.

In this pipeline, hydraulic and electrical noises have significant magnitudes compared with the measurements, and for this reason a fading memory average filter is used for all the measured signals Q_in, H_in, Q_out, H_out. For each such signal the filter is chosen as follows:

(31)

where denotes the filtered signal from s_i and is a filtering coefficient. The initial value for the friction estimate is taken from the reported one in data sheets and is shown in Table 1. The initial values of are the nominal flow values for the starting operating point and are calculated from the Darcy-Weisbach equation Mott (2005).

The values of the parameters used in the LDI algorithm are shown in Table 3. Comments about selection of many of the parameters presented in this table were given in the previous section. The results corresponding to one leak isolation experiment are reported hereafter.

Table 3: LDI Algorithm Parameters

Leak at 20 m. From the steady state shown in Table 4, the operating point (O.P.) changes at 200 s. This change is an increase in the pressure head, due to an increase in the pump speed. At 600 s a leak is started with an approximated magnitude of 0.00010 m³/s ( 2% of the nominal flow present in the pipeline just before the leak starts).

Table 4: Mean values of the measured and estimated quantities in experiment 1.

Figure 5a shows the measured and the estimated inlet and outlet flows. As it can be seen, when there is no leak the estimated and measured flows are similar. This shows that the model, the state estimator and the friction estimate of the LDI are adequate. As theory predicts, when the leak appears the measured inflow Q_in slightly increases and the measured outflow Q_out decreases. Meanwhile, their estimations and slightly decrease, since in a real setting when a leak appears the input pressure slightly decreases since there are no ideal pumps. Notice that since the observer is based on a leak-free model, it will in general give ≈ . In Fig. 5b the measured inlet and outlet heads (inputs) are presented. Notice that when the O.P. occurs, both heads increase and when the leak begins, both measurements decrease. This figure also shows that the noise magnitude is larger when the O.P. rises because when the pump speed increases the water turbulence is bigger. Fig. 6a shows that before the leak the residuals remain centered at zero and with a small magnitude, in such a way that the alarm is not activated in spite of the O.P. change, once the leak appears these values shift from zero. In Fig. 6b the reader may note that the cross correlation sum is insensitive to the O.P. change and that it is less sensitive to noise than the residuals, then the use of the cross correlation is more suitable to avoid false alarms. When the leak appears, the cross correlation significantly deviates from zero and it surpasses the threshold 24 s later, then the leak alarm is triggered. The friction estimate is shown in Fig. 6c: when the estimation process begins, the estimated value of ƒ quickly converges to the mean value of 0.02144. Once the O.P. changes, the value of ƒ decreases to the mean value of 0.02108. When the leak alarm is triggered, the friction value is frozen using Eq. (29) to a value of 0.02106. Figure 7a presents the leak magnitude estimation which has a mean value of 9.75 x 10^-5 m³/s, meaning an error of 2.5% compared with the real value of the leak magnitude. The estimation of the leak position is shown in Fig. 7b. The mean value of this estimation is 20,4 m with a standard deviation of 2.4 m, this estimation has an error of 0.5 % compared with the total length of the pipeline.

Figure 5: a) Measured flows and b) Pressure heads.

Figure 6: a) Estimated residuals, b) Cross correlation and c) Friction estimation.

Figure 7: a) Leak magnitude estimate and b) Leak position estimate.

From the presented experiment, it can be seen that acceptable isolation of small leaks has been obtained in spite of the intrinsic difficulties of the prototype, but in order to achieve this, the following practical considerations were taken into account:

Check the sensors calibration: The present algorithm is very sensitive to the initial calibration of the sensors, since small errors can produce false alarms or incorrect leak isolation.

Verify the sensors resolution: In a real setting the flow and pressure sensors resolution must be taken into account, since if a sensor is not able to detect small changes, then the effect caused by a small leak will not be detected. Thus, it is crucial to have the best possible resolution in sensors.

Decide if an adaptive friction estimation is needed: In this setting, and in other plastic pipelines, where a nonturbulent flow regime is present, it is essential to have an adaptive friction coefficient estimation as discussed along this paper.

Comments: a) The method of characteristics allows to derive a low dimensional model providing good accuracy, but it is necessary to satisfy Courant's condition, which imposes a restriction in the selection of the time and space steps, b) The calculated value of the pressure wave speed in this PCR pipeline allowed to select adequate step values, situation that in an equivalent short metallic pipeline could not be evident, since the wave speed value is greater (closer to the speed of sound in water). c) The obtained model leads to an estimator which can be easily implemented within an industrial controller, d) Due to the low roughness of this plastic pipeline, the flow regime present in this setting turns the friction coefficient sensitive to flow changes, therefore the adaptive estimation of ƒ becomes a crucial part of the LDI. d) Unlike in simulation works, where a fully known setting is manipulated, this work dealt with the intrinsic difficulties found in a real setting.

VI. CONCLUSIONS

In this paper, a well known Leak Detection and Isolation (LDI) algorithm, has been re-derived to use a model obtained from the Characteristic Lines Method. In order to achieve online leak isolation, it has been enhanced with an adaptive estimation of the friction coefficient and with a special freezing method. This algorithm has been implemented and evaluated on a plastic pipeline prototype transporting water.

In spite of the presence of unmodeled dynamics, high noise, friction coefficient variations and simplicity of the LDI algorithm, good detection and isolation of leaks, with a magnitude of 2% of the total flow, has been achieved with acceptable reliability and robustness to operating point changes in a real setting.

From the discussion along this paper, the presented LDI algorithm could be a good candidate for a field implementation. Concerning the future work, other LDI algorithms will be tested and new ones will be developed, with the aim of a future implementation on an a real aqueduct. Finally, we expect that this paper can become a useful reference for practitioners in this area.

APPENDIX

By using the method of characteristics to discretize model (l)-(2) when N = 2 is selected, the resulting discretized equations for node 1, 2 and 3 are:

	(32)
	(33)
	(34)

These equations can be written as a state variable representation, as follows:

(35)

where

ACKNOWLEDGMENTS
The authors thank the financial support of : Project S52974-Y and Project CB 81000 of CONACyT, Mexico and the French-Mexican Bilateral Projects CNRS-CONACyT J010/0159/10 and J100.322/2009

REFERENCES
1. API, Computational Pipeline Monitoring for Liquids, American Petroleum Institute RP 1130, 1st edition (2007).         [ Links ]
2. Begovich, O., A. Navarro, E.N. Sanchez and G. Besancon, "Comparison of two detection algorithms for pipeline leaks. " Proc. IEEE Int. Conf. Control Applications, 777-782 (2007).         [ Links ]
3. Begovich, O. and G. Valdovinos-Villalobos, "DSP application of a water-leak detection and isolation algorithm," Proc. 7th Int Electrical Engineering Computing Science and Automatic Control, 93-98 (2010).         [ Links ]
4. Besancon, G., D. Georges, O. Begovich, C. Verde and C. Aldana, "Direct observer design for leak detection and estimation in pipelines," Proceedings of the European Control Conference, Kos, Greece, 5666- 5670 (2007).         [ Links ]
5. Billmann, L. and R. Isermann, "Leak detection methods for pipelines," Automatica, 23, 381-385 (1987).         [ Links ]
6. Cinvestav, Pipeline prototype pictures at CINVESTAV-GDL, uri: http://www.gdl.cinvestav.mx/ofelia/(2010).         [ Links ]
7. Colombo, A.F., P. Lee and B.W. Karney, "A selective literature review of transient-based leak detection methods," Journal of Hydro-environment Research, 2, 212-227 (2009).         [ Links ]
8. Covas, D., H. Ramos and A.B. de Almeida, "Standing wave difference method for leak detection in pipeline systems," Journal of Hydraulic Engineering, 131, 1106-1116 (2005).         [ Links ]
9. Eiswirth, M. and L. Burn, "New methods for defect diagnosis of water pipelines," 4th Int. Conf. on Water Pipeline Systems, 28-30 (2001).         [ Links ]
10. Garcia-Tirado, J.F., B. Leon and O. Begovich, Validation of a semiphysical pipeline model for multi-leak diagnosis purposes, Proceeding of IAS TED International Symposium on Modelling and Simulation (2009).         [ Links ]
11. Kowalczuc, Z. and K. Gunawickrama, Fault Diagnosis: Models, Artificial Intelligence, Applications, Springer, 1st. edition (2004).         [ Links ]
12. Misiunas, D., J. Vtkovsky, G. Olsson, A. Simpson and M. Lambert, "Pipeline break detection using pressure transient monitoring." Journal of Water Resources Planning and Management, 131, 316-325 (2005).         [ Links ]
13. Mott, R.L., Applied Fluid Mechanics, Prentice Hall, 6th. edition (2005).         [ Links ]
14. Proakis, J.G. and D.G. Manolakis, Digital Signal Processing, Prentice Hall, 4th. edition (2006).         [ Links ]
15. Roberson, J.A., J.J. Casidy and M.H. Chaudhry, Hydraulic Engineering, John Wiley & Sons, 2nd. edition (1998).         [ Links ]
16. Scott, S.L. and M.A. Barrufet, Worldwide Assessment of Industry Leak Detection Capabilities for Single and Multiphase Pipelines, University of Texas, Austin (2003).         [ Links ]
17. Swamee, P.K. and A.K. Jain, "Explicit equations for pipe-flow problems," Journal of the Hydraulics Division, 102, 657-664 (1976).         [ Links ]
18. Torres, L., G. Besancon and D. Georges, "A collocation model for water-hammer dynamics with application to leak detection," Proc. IEEE Conf. Decision and Control, 3890-3894 (2008).         [ Links ]
19. UN/WWAP, UN World Water Development Report: Water for People, Water for Life UNESCO and Berghahn Books (2003).         [ Links ]
20. Verde, C., "Minimal order nonlinear observer for leak detection," Journal of Dynamic Systems, Measurement, and Control, 126, 467-472 (2004).         [ Links ]
21. Willoughby, D., Plastic Piping Handbook, McGraw-Hill Professional, 1st. edition (2001).         [ Links ]

Received: October 13, 2010.
Accepted: April 14, 2011.
Recommended by Subject Editor Jorge Solsona.