ABSTRACT
Pipeline leaks pose significant risks to industries, the environment, and individuals, so localizing pipeline leaks is crucial for enhancing pipeline safety during operation. This paper applies a method for localizing pipeline leakages, integrating transient signal detection with multi-sensor data fusion. Addressing the challenges in detecting small leaks amidst strong noise and uncertainty, the method employs the Dempster–Shafer evidence framework for data fusion and an algorithm to analyze transient pressure waves. Comparing it with three spectrum-based methods, the performance of the fusion method is discussed in the single-leakage and multi-leakage cases. In the single-leakage case, even with high levels of noise, the fusion algorithm delivers precise localization estimates. The fusion method excels over the other three methods in the multi-leakage case. The approach significantly enhances the accuracy of leak localization in water pipelines. Extensive simulations demonstrate the method's effectiveness, particularly in noisy environments, offering a promising solution for maintaining pipeline integrity and reducing resource wastage.
HIGHLIGHTS
The leak location pipeline transient model is obtained using transfer matrix analysis.
The method combines fusion algorithms with leak detection methods to improve detection accuracy.
This method has high noise immunity for small leakage detection.
INTRODUCTION
Pipeline transportation is a vital component of industrial production and everyday activities due to its numerous advantages, including high capacity, easy construction, cost-effectiveness, and straightforward operation (Moubayed et al. 2021). Despite these benefits, pipeline systems are susceptible to various challenges such as corrosion, ageing, and external forces (Huang et al. 2020). The occurrence of these issues is often influenced by both internal and external factors, leading to frequent incidents of pipeline leakage. The ramifications of these leaks extend beyond resource wastage, encompassing significant economic losses. For this reason, ensuring the safety and reliability of pipeline systems through prompt leak detection and localization is paramount in industrial operations and daily life.
In recent years, the field of leak detection and localization has seen a significant focus on transient-based techniques due to the promising and versatile nature of the fluid transient-based defect detection method. Extensive research efforts over the past few decades have contributed to the development of various commercially available leak detection techniques. These techniques range from simple physical detection methods to sophisticated acoustic approaches such as vibration signals (Kothandaraman et al. 2022), negative pressure waves (Chen et al. 2018; Lang 2021), and sound waves (Lang et al. 2018). The method for detecting defects using fluid transients involves introducing hydraulic waves into piping systems, measuring the pressure response at specific locations, and analyzing the signals to identify and characterize defects.
In practice, transient wave reflections from small leaks are typically weak and can be influenced by various uncertainties such as noise, traffic, mechanical equipment, and turbulence. To enhance the robustness of leak localization, two methods can be employed, as suggested by Keramat et al. (2019). The first approach, proposed by Wang et al. (2020a), involves conducting the experiment multiple times and using the average of several transient measurements. The wavelet transform technique has been shown to be effective in accurately pinpointing small leaks (Ferrante et al. 2007). However, conducting these transient experiments repeatedly is labor-intensive, disturbs the water supply, risks inducing pipeline structural fatigue, and is therefore impractical. The second solution suggests conducting the transient experiment just once but utilizing pressure signals from multiple sensors at various locations. The second solution proposes to perform only one transient experiment, using pressure signals from multiple sensors at different locations for leak detection by means of spectral analysis techniques. The utilization of sensors for health monitoring in contemporary urban pipe networks is on the rise (Qi et al. 2018). To enhance leak detection accuracy in noisy environments, the multi-sensor approach is gaining traction (Khaleghi et al. 2013; Wang et al. 2018). Leakage patterns are deterministic and manifest in each sensor's signal amidst random fluctuations.
Many applications, including sensor networks, automated control, and video/image processing, have been developed in the field of multi-sensor fusion. Researchers have used fuzzy set theory (Zadeh 1965), Dempster–Shafer evidence theory (DEST) (Shafer 1976), and probabilistic theory (Durrant-Whyte & Henderson 2008) to quantify data defects in information fusion. Probability distributions are used to represent data uncertainty, while fuzzy set theory is employed to represent data ambiguity. However, in cases that involve uncertain and ambiguous data, the DEST (Dempster 1967) is considered to be more general and flexible. In the context of leak localization problems, the pressure data obtained from each sensor often suffer from noise and uncertainty due to the system's complexity. Despite their imperfections, these pressure measurements provide valuable evidence about the presence and location of the leak. Modeling this evidence (Wang et al. 2020b) with probability distributions is difficult due to the unknown distribution of arbitrary uncertainty in the data. In contrast, transient wave data display a distinct structure. Hence, this study proposes a novel approach to dealing with uncertainty in transient wave data by utilizing the DEST framework.
Wang et al. (2019) used spectrum analysis for leak detection, but in the case of multiple leaks, the localization effect deteriorates and even misclassification occurs when the leaks are too close together. To solve this problem, we propose the DS-MUSIC-like method on top of this, which combines the DEST framework to analyze the transient waveform data to achieve multi-sensor fusion and reduce uncertainty. By applying the improved subspace-based DS-MUSIC-like algorithm to simulate small leaks by varying the leakage area, we are able to detect not only small leaks that are difficult to identify under strong noise but also leaks that are too close to each other successfully. This detection method takes location information into account and improves the accuracy and diagnostic capability of leak localization.
The paper is structured as follows: first, a description of the hydraulic transient model is presented. Then, the multi-source sensor fusion method is described in detail. Numerical simulations are performed to verify the effectiveness of the method. We present the results and discuss the performance of the DEST methodology for single and multiple leaks. Finally, conclusions are drawn.
BASIC THEORY
Transient pipeline modeling
and downstream tank location 
are specified. Denoting the sensor location near the downstream tank as 
, the leakage location is denoted by 
. 
Structure of the pipeline system under consideration.
In the pipe, the water pressure is represented by P and the flow rate by V. The water density is represented as 
, and the pressure wave velocity is denoted by a. 
represents the angle between the pipe and the horizontal plane, while g represents the gravitational acceleration. The Darcy–Weisbach friction factor is expressed as f, while the internal diameter of the pipe is indicated as D. The friction resistance term is linearized due to the second-order non-linearity of the turbulence pipe flow term, making direct conversion to the frequency domain difficult (Duan et al. 2018). In addition, time is represented by t, and the distance from the upstream extremity of the pipe is denoted by x.
and the head 
at 
and the discharge flow 
and the head 
at 
is determined by applying the transfer matrix technique to the value at 
, which is based on the linearized form of the frictional resistance term and the orifice equation.
is the stabilized leakage flow corresponding to the head 
, 
is the discharge coefficient of the leak, and 
is the flow area of the leak orifice.
.
is the size of the leak, and
is the pipe's elevation at the leak, and 
is additive unrelated Gaussian random disturbance with zero mean and covariance 
.
and 
, are assumed to be known. The upstream is connected to a reservoir, leading to the reasonable assumption that 
. The discharge 
can be estimated if a measurement station is located near the upstream boundary, denoted as 
. Assuming there is no leakage between 
and 
, the discharge at the upstream node 
can be determined using the pressure head measurement 
at 
. By utilizing the known boundary condition 
, it becomes possible to solve for 
.
represent the discrepancy in the pressure sensor readings before and after the leak, computed as
generalized model can be obtained as
is a vector with JM dimensions (Wang & Ghidaoui 2018):
This suggests that the head pressure difference value is linked to the leak's location, thus 
will be utilized subsequently.
MUSIC-like algorithm applied to leak detection
and aims to minimize the total power while imposing constraints on the desired signal power. The optimization problem associated with this algorithm (Zhang & Ng 2010) can be defined as follows:
The weight matrix 
represents the optimal solution vector of the problem. 
is an estimate of the received signal covariance vector. 
is a control parameter and 
represents a non-essential constant in the optimization process outcomes.
is the Lagrange multiplier.
Let 
represents the smallest eigenvalue, and 
represents the second smallest eigenvalue of the covariance matrix 
.
is the eigenvector that corresponds to the smallest eigenvalue of the matrix 
. The leakage location 
can be determined from peaks in the spatial power spectrum. The formula is expressed as
Multi-sensor leakage data fusion
Originally proposed by A.P. Dempster in 1967, the Dempster–Shafer theory of evidence, also referred to as the theory of belief function theory (Dempster 1967), was introduced to tackle multivalued mapping issues through the utilization of upper and lower bound probabilities. G. Shafer introduced the concept of belief function to develop the theory of evidence. He also created a set of mathematical methods for dealing with uncertainty reasoning, including evidence and combination. D–S theory is an extension of Bayesian reasoning that effectively captures the concept of uncertainty (Lin et al. 2021). It does not need to know the a priori probability and can be very useful.
represents the power set of Θ expressed as:
is quantized with the constraint that
The subset A of Θ is termed a focal element of m if m(A) is greater than zero, where m is any measure. The empty set is denoted by 
. The mass function m(A) indicates the level of support for A provided by the evidence.
, is merged using Dempster's rule for evidence combination (Shafer 1976):
as follows:
and the power set 
of 
. The mass function that describes the possible locations of leakage x is as follows:
in which 
stands for spectrum function.
Leak localization methods are summarized in Algorithm 1.
Algorithm 1. Localization of leaks through multi-sensor fusion within the Dempster–Shafer framework
SIMULATION RESULTS
Numerical setup
The validation of the proposed leak localization method using simulation data is presented in this section. Figure 1 displays the layout of the numerically simulated pipeline, with a valve located downstream of a single pipeline and pressure sensors positioned in the same downstream area. Another pressure sensor at 
is used to estimate 
. For downstream multi-sensor arrangements, there is a sensor ambiguity problem. When two sensors are too close to each other, the measurements received from the two sensors are linearly correlated. To avoid this problem in order to obtain the most leakage information, the distance between two sensors is required to be not less than 
(Wang 2021), where 
is the minimum wavelength of the detected wave.
The transfer matrix method is used to simulate unsteady wave propagation in the frequency range. It is assumed that pulse waves are generated when the valve rapidly closes and opens. The specified boundary conditions are 
and 
, with the simulation's key parameters listed in Table 1.
Parameters of the pipeline
| Piping parameters | Numerical value |
|---|---|
| Pipe length |  |
| Wave speed |  |
| Upstream reservoir head |  |
| Downstream reservoir head |  |
| Pipe diameter |  |
| Darcy–Weisbach coefficient |  |
| Steady-state discharge |  |
| Upstream boundary head |  |
| Downstream boundary discharge |  |
| Piping parameters | Numerical value |
|---|---|
| Pipe length | |
| Wave speed | |
| Upstream reservoir head | |
| Downstream reservoir head | |
| Pipe diameter | |
| Darcy–Weisbach coefficient | |
| Steady-state discharge | |
| Upstream boundary head | |
| Downstream boundary discharge | |
is the mean head difference and 
is the Gaussian white noise standard deviation.Leak localization performance with single leak and single sensor
and assuming the total leak parameter 
. The transient wave propagation is simulated with a sensor 
, and the DS fusion algorithm fuses the mass functions 
and 
obtained from 
and 
. For leakage detection, resonant and anti-resonant frequencies 
are used, where 
. The DS fusion algorithm is applied to a correlation 
matrix sample size of 620 (N = 620). To compare the DS fusion algorithm with other algorithms, the latter two belonging to the beamforming and subspace-based methods, respectively, the three beamforming algorithms are designed with different weighting vectors as candidate leakage position functions. The spatial power spectral function is maximized at the actual leakage location in these algorithms. Figure 2 shows the leakage localization results at a signal-to-noise ratio of 0 dB. From the leakage localization results in Figure 2, it can be observed that the Matched-Field Processing (MFP) method has a wider main flap with a large number of side lobes, particularly at 600, 800, and 1,200 m. In contrast, both Capon's beam-forming (BF) and MUSIC-like methods successfully achieve narrow peaks while retaining a relatively large amount of side lobes at 600 m. On the other hand, the DS-MUSIC-like method successfully achieves narrow peaks and eliminates all side lobes, outperforming these three methods. 
Detection of one leak utilizing four algorithms. The actual leak location is 1,400 m. Fused data sensor locations are 1,800 and 2,000 m. The SNR is 0 dB.
Detection of one leak utilizing four algorithms. The actual leak location is 1,400 m. Fused data sensor locations are 1,800 and 2,000 m. The SNR is 0 dB.
Leak localization performance with single leak and dual sensors
and leak location 
. The simulation of transient wave propagation is conducted with sensor locations 
and 
. The DS fusion algorithm fuses the mass function m1 obtained from 
and 
with the mass function 
obtained from 
and 
. The leak localization results in Figure 3 show that Capon BF, MUSIC-like, and DS-MUSIC-like methods all produce sharp peaks and reduce sidebands effectively, unlike the MFP method where the primary lobe is wider and the secondary lobe is significantly high around 1,600 m. All three methods outperform MFP in terms of suppressing side lobes and fluctuations. However, for relatively small signal-to-noise ratios (SNRs) (e.g. −40 dB in Figure 4), the performance of the algorithms deteriorates as the noise level increases. This is evident as all four algorithms produce results with side lobes while still being able to roughly locate the leak. Higher side lobes may be misidentified as leaks, compromising leak location accuracy, especially if the number of leaks is unknown. In terms of suppressing side lobes and fluctuations, the fusion method, DS-MUSIC-like, exhibits superior performance. 
Detection of one leak utilizing four algorithms. The actual leak location is 800 m. Fused data sensor locations are 1,800 and 2,000 m and 1,600 and 2,000 m. The SNR is 0 dB.
Detection of one leak utilizing four algorithms. The actual leak location is 800 m. Fused data sensor locations are 1,800 and 2,000 m and 1,600 and 2,000 m. The SNR is 0 dB.
Detection of one leak utilizing four algorithms. The actual leak location is 600 m. Fused data sensor locations are 1,800 and 2,000 m and 1,600 and 2,000 m. The SNR is −40 dB.
Detection of one leak utilizing four algorithms. The actual leak location is 600 m. Fused data sensor locations are 1,800 and 2,000 m and 1,600 and 2,000 m. The SNR is −40 dB.
is our simulated value, 
indicates the true leak location, and 
is the estimated value from the ith trial. The results shown in Figure 5 indicate that the RMSE of estimation decreases as SNR increases, with the DS-MUSIC-like algorithm exhibiting smaller errors compared with other methods. 
RMSE (m) of the first leak versus SNR.
Leak localization performance with dual leaks and single sensor
and 
with dimensions 
and 
, respectively. The sensor location is 
. Mass functions 
and 
obtained from 
and 
are fused by the DS fusion algorithm. Assuming a SNR of −30 dB for the measurement noise at all frequencies and a sample size N = 620, the results are displayed in Figure 6. Each figure shows a local maximum near each actual leak, confirming that all four methods accurately pinpoint two leaks. The first three algorithms, particularly the MFP algorithm, feature side flaps. The MFP algorithm is distinguished by its broad main flap and several side flaps, one of which is a high side flap at around 1,400 m. The wide main flap and the presence of high side flaps in the MFP algorithm can interfere with leak localization, especially when leak counts are unknown. In contrast, the DS fusion algorithm shows smooth curves in Figure 6 and proves to be effective in suppressing the side lobes. 
Localization of two leaks using four algorithms. Actual leak locations are 600 and 1,000 m. Fused data sensor locations are 1,600 and 2,000 m. The SNR is −30 dB.
Localization of two leaks using four algorithms. Actual leak locations are 600 and 1,000 m. Fused data sensor locations are 1,600 and 2,000 m. The SNR is −30 dB.
Leak localization performance with dual leaks and dual sensors
and 
with dimensions 
and 
, are displayed in Figure 7. The sensor locations are 
and 
. The DS fusion algorithm fuses the mass function 
obtained from 
and 
with the mass function 
obtained from 
and 
. Assuming a measurement noise SNR of −30 dB at all frequencies and a sample size of N = 620, it can be observed that each method exhibits different degrees of side lobes and higher side lobes at certain locations which may impact the determination of the leakage location. In comparison to the aforementioned methods, the DS-MUSIC-like algorithm effectively suppresses most of the side lobes through data fusion and accurately locates the leaks. 
Localization of two leaks using four algorithms. Actual leak locations are 600 and 1,400 m. Fused data sensor locations are 1,600 and 2,000 m and 1,700 and 2,000 m. The SNR is −30 dB.
Localization of two leaks using four algorithms. Actual leak locations are 600 and 1,400 m. Fused data sensor locations are 1,600 and 2,000 m and 1,700 and 2,000 m. The SNR is −30 dB.
In the DS-MUSIC-like algorithm, the leakage probability of the side flap is less than 0.2, and it can be considered a leakage-free case. Compared with other methods, the fusion algorithm greatly suppresses the generation of side flaps and reduces the leakage detection error.
CONCLUSIONS
This study proposes a method to improve leak location accuracy by introducing multi-sensor measurements is proposed. The information leakage, as measured by multiple sensors, is extracted and fused using the DEST framework. It then uses a multi-sensor leak location analysis method. Uncertainty noise and transient wave measurements are used to help solve the problem of locating leaks in pipelines. The simulated results show that the method efficiently combines multi-sensor information. The method not only detects small leaks that are difficult to identify in strong noise but also successfully identifies leaks when the distance is too close, improving positioning accuracy and precision.
In practical applications, the distribution of uncertainties remains elusive due to a dearth of understanding of system architecture and equipment dynamics, inaccuracies in numerical computations and modeling, disturbances from traffic and other external factors, as well as imprecise measurements pertaining to wave speed, friction factor, and steady-state discharge. Consequently, advancing this field necessitates a thorough examination of the diverse uncertainties that could potentially impact leak detection. Furthermore, the simulation experiments presented in this paper have only encompassed scenarios with one or two leaks. In the case of a larger number of leaks, the optimization problem has to be solved. Therefore, in future studies, other techniques will be considered to reduce computational complexity and cost.
FUNDING
This work was supported by the Science and Technology Development Plan Project of Jilin Province, China (Grant No. 20230201068GX).
DATA AVAILABILITY STATEMENT
Data cannot be made publicly available; readers should contact the corresponding author for details.
CONFLICT OF INTEREST
The authors declare there is no conflict.
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