## Abstract

Water pipe networks have a large number of branch joints. Branch joint shunting generates vortices in the fluid, which excite the pipe wall to produce a type of branch noise. The branch noise is coupled with the leak source signal through the pipe. Here, a novel leak location protocol based on the complex-valued FastICA method (C-FastICA) is proposed to address the leak location problem under the branch noise interference. The C-FastICA, a complex-value domain blind deconvolution algorithm, effectively extended the cost function, constraint function, and iteration rules of the instantaneous linear FastICA into the complex-valued domain. The C-FastICA method was used to realize the separation of branch noise and leak source signal. The experimental results showed that the separation efficiency of the C-FastICA was higher than that of time-domain blind convolution separation (T-BCS). Furthermore, the relative location error of the C-FastICA method to the leak point was less than 14.238%, which was significantly lower than in traditional T-BCS and direct cross-correlation (DCC) technology.

## HIGHLIGHTS

The paper proposed a procedure based on the complex-valued FastICA method (C-FastICA) to address the leak location problem under the branch noise interference.

The accuracy of the proposed C-FastICA leak location method is higher than the traditional cross-correlation location technology and time-domain deconvolution leak location technology.

## INTRODUCTION

Leaks in water supply pipelines not only cause economic losses and environmental pollution, but also endanger the health and safety of urban residents. The main causes of pipe leaks are pipeline aging, chemical corrosion, external force damage, and management and maintenance negligence. According to the ‘In-depth Investigation and Investment Prospect Analysis Report of China's Urban Water Supply Industry for 2020–2026’ (Zhiyan Consulting 2019), the annual average rate of pipe leak in mainland China is 15.3% and some cities even exceed 25%, which is higher than the standard in many developed countries.

The acoustic emission (AE) detection method (Kim & Lee 2009; Brennan *et al.* 2016; Gao *et al.* 2017; Almeida *et al.* 2018; Kassab *et al.* 2019; Xiao *et al.* 2020) is widely used in detection of pipe leaks due to its low cost, non-intrusiveness, and high efficiency. The classic AE pipe detection is shown in Figure 1; the acoustic or vibration sensors are closely attached to both ends of the leak pipe to capture the leak signals. However, in real-world leak detection, various external and internal noises may interfere with the AE leak signal. External noises include those generated by automobiles, machines, and humans, whereas the internal noises are mainly caused by branch joints, elbow joints, valve joints, and other pipe joints. To effectively remove noise interference, Ahadi & Bakhtiar (2010) proposed a new method that captures the leak sound signal-signature in time domain based on the Short Time Fourier Transforms (STFT). The leak point was then located in the time-frequency domain using the tuned wavelet method. In a different study by Ting *et al.* (2021), a shift-invariant Dual Tree Complex Wavelet Transform (DTCWT) was used to remove noise interference in water pipe leak locations. Jafari *et al.* (2020) used an extension version of Kalman filter observer to detect and locate the blockage in the pipeline. Saqib *et al.* (2017) used a wavelet-based adaptive thresholding scheme to restrain the noise inter-ference within the AE leak signals. Taghvaei *et al.* (2006) used a cepstrum analysis to detect and locate leaks in pipeline networks.

Butterfield *et al.* (2017) investigated the correlation between the leak flow rate and signal processing techniques such as the Vibro-Acoustic Emission (VAE) counts, signal Root Mean Square (RMS), peak in magnitude of the power spectral density, and octave banding. The results showed a strong correlation between the RMS and leak flow rate, prompting the development of a flow prediction model that was based on the RMS parameters. Subsequently, Mostafapour & Davoudi (2013) used the Donnell's non-linear theory, Galerkin method, and Runge–Kutta numerical method to develop an AE model induced by pipe leak vibration. Yu & Li (2017) proposed an experimental investigation into the detection of minor leaks. The time-domain waves, frequency, and energy features of the AE signals were first extracted and compared, before applying the SVM to recognize the minor leaks in the extracted features. In general, these de-noise methods are the procedure of removing or suppressing additive noise. However, these methods have two drawbacks. Firstly, environmental variables in actual leak detection conditions impact the mathematical models, interfering with the task and potentially leading to premature completion. Secondly, these methods generally capture leak signals from a straight leak pipe that is devoid of branch joints, and they consider mixed leak signals to be equivalent to the superposition of noise and leak sources.

In leak conditions, the mixed sound signal propagates along the pipe wall after the coupling and superimposing of pipe noise and the leak sound signal. Since the propagation mechanism is a convolutional process, the leak signal containing the pipe noise should be separated using a deconvolution method. All the noise generated by the pipe shape, including the noise caused by the branch flow, is a type of posterior noise with unknown time domain, frequency domain, and probability density characteristics. Therefore, the above-mentioned methods are mainly based on power spectrum characteristics, time-frequency characteristics, and other prior noise information that can be established or hypothesized to realize the AE signal processing analysis. Accordingly, the highlighted methods and techniques cannot be directly applied to eliminate branch noise because of the limited amount of prior knowledge of branch type leak detection. In addition, the branch noise and leak source signal have a convolution relationship with the propagation channel (pipe wall). Therefore, isolating the leak source signal from the branch-noise containing leak signal is a typical blind deconvolution separation (BSS) problem (Sun *et al.* 2015). Without knowing the probability density characteristic, time-domain and frequency-domain characteristics of the leak source signal and the branch noise signal, the blind separation algorithm can still complete the separation of the observation signal (leak mixing signal) obtained by the sound sensor.

In blind separation, FastICA is widely used due to its fast convergence and high separation precision. Therefore, to efficiently extract the leak source signal from the mixed leak signal containing branch noise, this paper proposes a leak detection procedure that is based on the complex-valued FastICA method (C-FastICA). As an improved algorithm of FastICA, C-FastICA effectively extends the cost function, constraint function, and iteration rules of FastICA to the complex-valued domain. In section two of this paper, a computational fluid dynamics (CFD) example is used to describe the current problem in leak location of branch pipe. Section three briefly introduces the C-FastICA theory. Section four covers the leak location experiment, and conclusions are made in section five.

## PROBLEM DESCRIPTION

The branch joint that connects a branch pipe to the main pipeline system changes the flow state of the fluid thus generating water eddies near the branch joint. The eddies excite the pipe wall, resulting in a kind of vortex-induced noise referred to in this paper as branch noise. This kind of branch noise has unsteady-state characteristics and can be regarded as an unsteady-state acoustic response caused by branch joint (Wang *et al.* 2005; Meniconi *et al.* 2021).

In this section, the computational fluid dynamics (CFD) software from ANSYS Fluent was used to simulate the eddy phenomenon in leak pipe with a branch joint (Figure 2). In preparatory work, the CAD model of leak pipe was built using SolidWorks 2014 software. The three-dimensional mesh was then generated using the ICEM software based on the CAD pipe model. The Fluent software provides a transient large eddy simulation (LES) solver. In calculation, a LES model was selected to simulate the steady flow field. The simulation was performed on a server called Dell Precision T7600 with an Intel Xeon (R) CPU E5-2678 W 0 at 3.10 GHz (16 CPUs) and 128 GB RAM. After the simulated calculation reached a steady state, we used CFD-Post software to post-process the calculation results.

Figure 3 shows the distribution of vortex core near the branch joint at four different moments: t = 500 steps, t = 1000 steps, t = 1500 steps, and t = 2000 steps. Figure 3 shows that as the time step increases, the maximum of fluid velocity displays a decreasing trend. According to the definition of vorticity in , where is vorticity of vortex core, is the Hamiltonian, is the rotational velocity of the fluid, the vorticity is positively correlated with the fluid velocity, and the decrease in velocity leads to a decrease in vorticity. Due to the existence of this dynamically vortex core changing (as shown within the small black rectangle), the tube wall is excited to vibrate and to form vortex-induced noise, more details about vortex-induced noise are described in the literature (Chu 2006; Zhang *et al.* 2015; Zhang *et al.* 2016). This vortex-induced noise is mixed with the leak signal source and collected by a sensor installed on the pipeline, which becomes a major interference source for leak detection. Thus, we inferred that the leak detection will inevitably be interfered with due to the existence of one or more branch joints.

## THE LEAK LOCATION METHOD BASED ON C-FastICA ALGORITHM

### Building convolution model for leak

*et al.*2021; Deep

*et al.*2022). The mathematical analysis formula of the leak model is as follows:where and are observation matrices acquired from two sensors; and are two different transfer matrices; is the unknown mixing source matrix, ; is leak source; is branch noise; and and are the random interference factors. To further simplify Equation (1), we transformed its time domain to frequency domain, resulting in the product form of Equation (1):

In the frequency domain, the convolutional mixing model was transformed into a linear instantaneous mixing model, and the C-FastICA was used to separate the mixed signal. Following the completion of the calculations, the frequency domain results were converted into time-domain sources, as shown in Figure 4.

### Background of the C-FastICA method

**is the source signal,**

*s***is the real part of**

*s*_{r}**, and**

*s***is the imaginary part of**

*s*_{i}**;**

*s***is the mixed signal,**

*x***is the real part of**

*x*_{r}**, and**

*x***is the imaginary part of**

*x*_{i}**;**

*x***is the unmixing matrix, and**

*W***is the real part of**

*W*_{r}**,**

*W***is the imaginary part of**

*W*_{i}**. According to the Kuhn-Tucker conditions (Lv**

*W**et al.*2007), under the constraint condition , the maximum cost function was obtained by the first order derivative of the following formula:where is the nonlinear function, it is defined as . Next, we differentiated the real and imaginary parts of

**in Equation (5). The first term on the left side of Equation (5) can be written as:**

*W***was obtained as follows:**

*W***:where is the frequency-domain output estimation, the is frequency-domain source signal,**

*x*

*W**is the frequency-domain unmixing matrix. After the calculation of Equation (12), we transformed the complex-valued output into the time domain:where is the time-domain output estimation, the proposed C-FastICA is summarized in Algorithm 1.*

_{f}Algorithm 1: Complex- valued FastICA (C-FastICA) |

(1). The time-domain observation signal is transformed to complex-valued form through FFT: ; |

(2). Initialize ; |

(3). According to the definition , calculate the and ; |

(4). Initialize using Equation (12); |

(5). For iter ≥ 0: |

Calculate |

, |

(6). End for convergence:. |

(7). Cross-correlation estimation: |

(8). Leak location: |

Algorithm 1: Complex- valued FastICA (C-FastICA) |

(1). The time-domain observation signal is transformed to complex-valued form through FFT: ; |

(2). Initialize ; |

(3). According to the definition , calculate the and ; |

(4). Initialize using Equation (12); |

(5). For iter ≥ 0: |

Calculate |

, |

(6). End for convergence:. |

(7). Cross-correlation estimation: |

(8). Leak location: |

## LEAK LOCATION EXPERIMENT

The main leak location steps are shown in Figure 5, and are listed below as used in the experiment:

*Step 1*: two mixing leak signal *x*_{1}(n) and *x*_{2}(n) were acquired by acceleration sensors.

*Step 2*: the obtained *x*_{1}(n) and *x*_{2}(n) were transformed into frequency domain, then, two frequency-domain signals *x*_{1}(*f*) and *x*_{2}(*f*) were obtained.

*Step 3*: the *x*_{1}(*f*) and *x*_{2}(*f*) were treated as input for C-FastICA separation, and the unmixing matrixes *W*_{1}(*f*) and *W*_{2}(*f*) were calculated.

*Step 4*: we further computed the output *y*_{1}(*f*) and *y*_{2}(*f*) based on the unmixing matrixes *W*_{1}(*f*) and *W*_{2}(*f*).

*Step 5*: the output *y*_{1}(*f*) and *y*_{2}(*f*) were transformed into time domain, then, the time-domain output *y*_{1}(*t*) and *y*_{2}(*t*) were obtained.

*Step 6*: the cross-correlation method was used to obtain the time delay estimation (TDE) based on the obtained *y*_{1}(*t*) and *y*_{2}(*t*) in step 5.

*Step 7*: we combined the known distance *L* = 70 m, TDE, and the empirical leak sound velocity *c* = 1495 m/s to complete the leak location task.

### Conditions of the experiment

In this section, we used the acceleration sensors to collect two mixing leak signals containing branch noise, which were then separated using the C-FastICA algorithm. We selected a cast iron water supply pipe with a diameter of 100 mm and a test length of 70 m for the experiment. The branch node has water consumption in the corresponding branch pipeline. The pipe inner pressure was 0.6 MPa, and the inlet flow rate was approximately 2 m/s. The simulated leak point, which was 42 m away from sensor 1, was replaced by a small valve (leakage aperture = 15 mm) and the data sampling rate was set at 10 KHz. More details on the used parameters are shown in Table 1. A schematic diagram of the leak location experiment (Figure 6), and real diagrams of the two sensors and branch joints (Figure 7) are as shown below. Two sensors were placed at the different ends of the main pipe, and the collected leak signals were wirelessly transmitted to the host computer (Figure 6).

Parameters . | Values . |
---|---|

Pipe diameter | 100 mm |

Leak valve diameter | 15 mm |

Test pipe length: L + _{1}L + _{2}L _{3} | 70 m |

L: Sensor 1 distance from the leak point _{1} | L = 42 m _{1} |

L: Branch joint distance from the leak point _{2} | L = 8 m _{2} |

L: Sensor 2 distance from the branch joint _{3} | L = 20 m _{3} |

Pipe inner pressure | 0.6 MPa |

Flow velocity | 2 m/s |

Sampling rate | 10 KHz |

Parameters . | Values . |
---|---|

Pipe diameter | 100 mm |

Leak valve diameter | 15 mm |

Test pipe length: L + _{1}L + _{2}L _{3} | 70 m |

L: Sensor 1 distance from the leak point _{1} | L = 42 m _{1} |

L: Branch joint distance from the leak point _{2} | L = 8 m _{2} |

L: Sensor 2 distance from the branch joint _{3} | L = 20 m _{3} |

Pipe inner pressure | 0.6 MPa |

Flow velocity | 2 m/s |

Sampling rate | 10 KHz |

### Leak location processing

We chose the leak signal collected by Sensor 1 as an example, and used the C-FastICA algorithm to process it. The time-domain (Figure 8(a)) and frequency-domain (Figure 8(b)) forms of the leak signal containing branch noise are shown below.

To prove that the C-FastICA algorithm outperforms the traditional algorithm, we separated the mixing leak signal using the well-known time-domain blind convolution separation (T-BCS) algorithm. The signal had to be preprocessed before separating the mixing leak signal to suppress the influence of the additive noise *n*_{1} and *n*_{2} in Equation (1). After removing the additive noise in the mixing leak signal, the leak acoustic source and branch noise are the main components in the mixing leak signal. Then, the C-FastICA was used to perform blind separation for the mixing leak signal in the frequency domain. For more information on suppressing the additive noise, see Bell & Sejnowski (1995). The C-FastICA algorithm revealed that the spectral components of *y*_{1} and *y*_{2} were mainly occupied in the range of 1700–2600 and 1100–4000 Hz, respectively (Figure 9(a)). The T-BCS algorithm results indicated that the spectral characteristics of *y*_{1} and *y*_{2} were between 1600–2200 and 1000–4000 Hz, respectively (Figure 9(b)). This denotes that *y*_{2} has a wider frequency bandwidth than *y*_{1}. Therefore, we can infer from the previous leak theory(Diao *et al.* 2020; Scussel *et al.* 2021) that the randomness of the leak signal is often greater than that of other stable noise signals, except white noise, implying that the bandwidth of the leak source signal is wider than other noise signals. Consequently, we concluded that *y*_{2} was the leak source signal obtained by the separation algorithm, and *y*_{1} was the branch noise. When the number of branch pipe nodes is more than one, there will be multiple narrow frequency components in the separation result, and these narrow frequency components are considered branch noise.

*i*,

*j*)th element in the global system matrix

**. The algorithm is more convergent when the ISI output curve of each channel is closer to zero. The ISI output curves indicated that each of the two channels converged at about 300 steps in C-FastICA (Figure 10(a)) and at about 400 steps in T-BCS (Figure 10(b)). Thus, convergence is faster in the C-FastICA algorithm than in T-BCS arigorithm.**

*C**PI*was introduced to compare the separated degree of the C-FastICA and the T-BCS. The closer the

*PI*value is to zero, the higher the separation degree of the mixed signal. The

*PI*was defined as follows (Leung & Siu 2007):where

*M*is the number of source components,

*c*is the element of

_{ik}**=**

*C***·**

*W***,**

*A***is the mixing matrix, and**

*A***is the separation matrix obtained by separation algorithms. The C-FastICA algorithm and the T-BCS algorithm were used to separate the leak signal containing the branch noise for another 50 times. It should be emphasized that the leak mixing signal and other specific experimental parameters used in these 50 times experiments remain the same. The**

*W**PI*results are shown in Figure 11.

*PI*coefficients of C-FastICA algorithm were smaller, with a median value of about 0.192, than those of the T-BCS algorithm which had a median value of about 0.237 (Figure 11). Therefore, C-FastICA had a higher separation degree for the mixing leak signals, and the obtained leak source contained more leak characteristics. Following the separation calculation for the mixing leak signal collected by Sensor 1, we processed the mixing leak signal collected by Sensor 2 in the same way. The cross-correlation method was then used to obtain the TDE

*τ*based on the two leak source signals:where and are two leak sources obtained by blind separation, is from Sensor 1, is from Sensor 2, is the delay parameter. The leak hole location was computed using the distance

*L*= 42 m, TDE

_{1}*τ,*and the known leak sound velocity

*c*= 1495 m/s as follows:where

*L*is equivalent to . The peak value obtained using the C-FastICA algorithm (Figure 12(a)) was the closest to the true leak location compared to peak values shown by the T-BCS algorithm (Figure 12(b)) and the direct cross-correlation method (DCC; Figure 12(c)).

Since the branch noise was not removed before the analysis using the DCC method, it had a significant impact on the cross-correlation estimation thus yielding a leak location value with the largest error (Figure 12(c)) compared to the other two methods. To further verify that the separation performance of C-FastICA for mixed leak signals was better than T-BCS, this experiment was replicated eight times. The leak location results for each method are shown in Table 2. In the eight times leak location experiments in Table 2, eight acquisitions of leak mixing signal were carried out, the leak mixing signal used in each experiment is different.

Number . | C-FastICA . | T-BCS . | DCC . | |||
---|---|---|---|---|---|---|

Distance (m) . | . | Distance (m) . | . | Distance (m) . | . | |

1 | 36.82 | 12.333 | 35.71 | 14.976 | 61.24 | 45.810 |

2 | 36.95 | 12.024 | 34.07 | 18.881 | NaN | NaN |

3 | 37.05 | 11.786 | 34.59 | 17.643 | NaN | NaN |

4 | 37.54 | 10.619 | 32.82 | 21.857 | 58.96 | 40.381 |

5 | 37.12 | 11.619 | 33.29 | 20.738 | 66.53 | 58.405 |

6 | 36.76 | 12.476 | 35.17 | 16.262 | NaN | NaN |

7 | 36.02 | 14.238 | 34.69 | 17.405 | NaN | NaN |

8 | 37.33 | 11.119 | 34.18 | 18.619 | 56.82 | 35.286 |

9 | 37.98 | 9.571 | 35.02 | 16.619 | NaN | NaN |

Number . | C-FastICA . | T-BCS . | DCC . | |||
---|---|---|---|---|---|---|

Distance (m) . | . | Distance (m) . | . | Distance (m) . | . | |

1 | 36.82 | 12.333 | 35.71 | 14.976 | 61.24 | 45.810 |

2 | 36.95 | 12.024 | 34.07 | 18.881 | NaN | NaN |

3 | 37.05 | 11.786 | 34.59 | 17.643 | NaN | NaN |

4 | 37.54 | 10.619 | 32.82 | 21.857 | 58.96 | 40.381 |

5 | 37.12 | 11.619 | 33.29 | 20.738 | 66.53 | 58.405 |

6 | 36.76 | 12.476 | 35.17 | 16.262 | NaN | NaN |

7 | 36.02 | 14.238 | 34.69 | 17.405 | NaN | NaN |

8 | 37.33 | 11.119 | 34.18 | 18.619 | 56.82 | 35.286 |

9 | 37.98 | 9.571 | 35.02 | 16.619 | NaN | NaN |

The relative location error (*δ*) of C-FastICA did not exceed 14.238%, while in the T-BCS algorithm it reached 21.857%, indicating that C-FastICA algorithm had a higher position accuracy than T-BCS algorithm. However, noise had a significant impact on the DCC method whose relative error value was greater than 58%, and the leak location could not even be completed in numbers 2, 3, 6, 7, and 9.

## CONCLUSION

This paper proposed the use of the C-FastICA method to separate the mixing leak signal and to solve the branch noise interference problem in the leak location of branch water pipes. Following the separation calculation, the leak sources were identified, and the leak location was completed by combining the prior leak sound velocity and the known pipe length. The feasibility of the proposed leak location method was proved through experiments, and the following conclusions were reached:

- (1)
When there was a branch joint in a water pipeline, the leak signal was mixed with the branch noise, and the T-BCS and DCC methods for locating the leak point had a significant error or were unable to complete the location. It shows that the influence of the branch noise cannot be ignored on the branch pipe leak location.

- (2)
The experiments proved that the C-FastICA algorithm has a faster convergence speed and a higher separation level for the mixed leak signals than the T-BCS algorithm.

## ACKNOWLEDGEMENTS

This project was supported by the National Natural Science Foundation of China (No. 51675069), the National Natural Science Foundation of China (No. 51775070), and the Fundamental Research Funds for the Central Universities (Nos. 2018CDQYGD0020, cqu2018CDHB1A05).

## DATA AVAILABILITY STATEMENT

Data cannot be made publicly available; readers should contact the corresponding author for details.

## REFERENCES

## Author notes

These authors contributed equally to this work.