Multi-Stage Burst Localization Based on Spatio-Temporal Information Analysis for District Metered Areas in Water Distribution Networks

Abstract

Burst events in Water Distribution Networks (WDNs) pose a significant threat to the safety of water supply, leading people to focus on efficient methods for burst localization and prompt repair. This paper proposes a multi-stage burst localization method, which includes preliminary region determination and precise localization analysis. Based on the hydraulic model and spatio-temporal information, the effective sensor sequences and monitoring areas of the nodes are determined. In the first stage, the preliminary burst region is determined based on the monitoring region of sensors and the alarm sensors. In the second stage, localization metrics are used to analyze the dissimilarity degree between burst data from the hydraulic model and the monitoring data from the effective sensors at each node. This analysis helps identify candidate burst nodes and determine their localization priorities. The localization model is tested on the C-Town network to obtain comparative results. The method effectively reduces the burst region, minimizes the search region, and significantly improves the efficiency of burst localization. For precise localization, it accurately localizes the burst event by prioritizing the possibilities of the burst location.

1. Introduction

Burst events in Water Distribution Networks (WDNs) can result in significant economic losses during the transportation of water to consumers, leading to increased WDN repair costs and, ultimately, additional costs to end consumers [1]. In many WDNs, the leak losses are estimated to account for up to 30% of the total water withdrawn [2]. Given the rapid development of urban areas and increasing demand for water supply, this percentage holds significant importance [3,4]. In addition, burst reduction is a challenging task due to the complexity of WDNs [5,6]. In recent years, the approach of using District Metered Areas (DMAs) as units for control and management has been widely adopted to address leakage issues in WDNs [7]. However, quickly locating and repairing bursts within DMAs remain a significant challenge, due to factors such as external noise and fluctuations in consumer water usage. Therefore, burst detection and localization approaches in DMAs have been extensively researched [8].

Burst events entail unawareness, detection, localization, and repair phases. Burst detection and localization are two essential components of burst monitoring. A challenge associated with burst localization is the identification of the exact location of a burst in WDNs using data from deployed sensors. Assuming the efficient operation of the burst detection system and activation of the burst alarm, the solution to the burst localization problem depends on the verification of the existence of a burst before proceeding with the localization process. Only after the existence of a burst event is confirmed can the burst localization be performed. The goal of the localization process is to select nodes or areas that are as close as possible to the actual burst location.

In previous studies, the scholars have proposed various methods for burst detection and localization. The techniques can be broadly categorized into the following main types: hardware-based methods, transient-based methods, steady-state hydraulic model-based methods, statistical methods, and data-driven methods [9,10]. Hardware-based methods primarily rely on specialized devices to determine the occurrence and specific location of the bursts [9]. The devices include leak noise loggers, listening rods, ground-penetrating radar, gas injection, and so on. Monitoring methods based on hardware devices are not suitable for long-term continuous real-time monitoring. They are typically used for periodic inspections. Implementing monitoring using this method often requires a significant investment in manpower and incurs substantial equipment costs. The monitoring range is limited, making it suitable for small-scale monitoring but not for a widespread and extensive use [11].

The principle of burst localization based on transient signals is based on the propagation, attenuation, and reflection of transient pressure waves generated by sudden burst events in the network [12,13,14,15,16,17,18]. The typical process of this method involves introducing pressure wave events, comparing measured and simulated results, and identifying burst events based on time–frequency domain analysis. However, the transient wave signals used as part of this method are susceptible to interference from other noise sources. Currently, this method has only been validated in laboratory pipelines and controlled environments and has not been applied to practical network settings [19].

Researchers have shown a significant interest in methods based on steady-state hydraulic models, owing to the widespread adoption of Supervisory Control and Data Acquisition (SCADA) systems. The model-based method involves analyzing the correlation between the measured values from sensors and the estimates based on the calibrated model [20,21]. Furthermore, the method can be classified into three types: sensitivity analysis, optimization calibration, and error domain model falsification [22]. Over time, advances in modeling and simulation software tools have spurred interest in these methodologies, resulting in a myriad of papers in the literature. The method compares burst and normal scenarios to determine the exact location of the leak [23]. To handle uncertainty, the method incorporates the Dempster–Shafer reasoning, which ensures robustness in the burst detection process. This correlation analysis is used to determine the occurrence of burst events and their locations. The difference between the monitored pressure in burst and normal conditions based on the hydraulic model is used to identify the pipe with the greatest influence on each sensor, thereby determining candidate burst areas [24]. The reliability of this method largely depends on the availability and accuracy of hydraulic models.

Statistical Process Control (SPC) charts with a set of control limits provide intuitive and cost-effective tools for monitoring and displaying the unusual behavior of a process. An Exponential Weighted Moving Average (EWMA) model has been used to detect a leak event in a DMA equipped with Automated Meter Reading (AMR), and the detection time has been estimated at approximately 1 week after its occurrence [25]. In a subsequent study, a hybrid method has been used to combine the results of WEC regulations and the cumulative sum (CUSUM) method, although the Detection Time (DT) for burst events remained at least 5 h. As a result, the accuracy and practicality of traditional SPC methods remain a challenge [26]. The data-driven methods use machine learning techniques to interpret the collected data for burst detection and localization. Support Vector Machines (SVMs) [27], Artificial Neural Networks (ANNs), and Random Forests (RFs) [28] have been employed, for instance, in the identification of burst signals. Recently, a study has been able to improve the dense convolutional network by replacing the convolutional layers with linear connections in the deep learning algorithm [29]. They used a large amount of simulated monitoring data from pressure sensors with burst and normal labels to train the improved fully linear DenseNet for burst localization. A real-time burst localization method using time delay neural networks and flow or pressure measurements has also been presented [30], as well as a method combining artificial neural networks for pipeline burst diagnosis using pressure and flow data [31]. The method is suitable for WDNs that have a large number of sensors, but it is not well-suited for complex and large-scale WDNs. Then, a significant amount of high-quality burst data at specific timestamps are required to achieve high accuracy. However, currently, there is a lack of precise timestamp data for each occurrence and end of burst events. Missing data/outliers caused by intermittent sensor failures and limited battery power have become common features of sensor networks. These instabilities severely limit the practical application of the above methods.

In addition, in recent years, there have been studies that have complemented the hydraulic model and statistical methods, both of which have their own advantages and limitations, and have jointly overcome the uncertainty of node requirements and the subjectivity of manually defining the burst detection threshold. A two-stage leakage detection model has been proposed [32]. In the first stage, burst events are detected through statistical methods of data classification and feature extraction. Then, in the second stage, the hydraulic model of L-town WDNs is calibrated and the leakage location determined using experimental optimization methods.

Although existing studies have investigated the burst localization, there are still some gaps with respect to certain aspects. In most of the abovementioned studies, the burst localization issue analyzes the residual sensitivity and estimates the burst locations using residual values [33]. It requires exploring various burst flow rate conditions for each node in the WDNs. This leads to a large search space, resulting in inefficiency and time consumption. This traditional model-based method, which directly compares residuals, overlooks the fact that each sensor has a specific range of monitoring nodes. During the localization process, area divisions can be made based on the sensors to achieve the purpose of narrowing down the scope.

The pressure data monitored by the SCADA system are a classic timeseries, and there is a certain spatio-temporal correlation among the monitoring data of sensor networks in WDNs. In addition, the pressure data from different sensors are typically spatially correlated. In particular, assuming that a burst occurs at a location, at least one pressure sensor will show synchronous pressure changes. By effectively utilizing the spatio-temporal correlation information from multiple sensors, the uncertainty of individual sensors can be significantly reduced, thereby enhancing the robustness of the algorithm. Different network topologies, varied locations of pipe bursts, and different flow rates from burst pipes can all affect the spatio-temporal correlation of sensor information. Burst localization has long been relying on monitoring information from either a single sensor or all sensors in WDNs without selectively using the data based on the importance of different sensors. On one hand, relying on individual pressure sensor for localization results in lower accuracy and an inability to achieve accurate localization [34,35]. On the other hand, when the data from all sensors are used for burst localization, the accuracy of the localization method is often affected by measurement noise interference from ineffective sensors, leading to a lower accuracy [27,36].

To address the above problems, we propose a novel multi-stage model-based burst localization framework based on spatio-temporal information from multiple sensors at multiple timesteps. The main contributions are as follows. First, in order to take full advantage of the spatio-temporal correlation between burst locations and different sensors, the monitoring region of the sensor and the Effective Sensor Sequences (ESS) of the nodes are proposed. Secondly, in order to improve localization efficiency, the multi-stage localization approach significantly reduces the scope of localization by effectively excluding normal positions of non-burst pipes. Thirdly, in order to address the issue of low localization accuracy, a localization indicator based on continuous temporal dissimilarity is proposed for accurate localization. The burst locations are then ranked based on the likelihood of occurrence, providing effective guidance for subsequent maintenance work by water utilities. Taking the C-town network as an example, the effectiveness of different dataset verification methods is validated.

2. Methods

2.1. Overview

The burst detection method [37] is used to determine if a burst has occurred. If no sensor alarm is triggered, the network status is considered to be normal. Conversely, if a sensor alarm is triggered, the next step is to locate the burst and perform repairs immediately. Since the burst detection method has been studied in the literature [37], our focus is on burst localization once the detection alarm has been triggered.

Figure 1 illustrates the method framework proposed in this paper, consisting of the following steps. By analyzing the pressure sensitivity matrix simulated based on the hydraulic model, and by utilizing the spatial correlation between sensors and bursts, this study first develops the concept of monitoring regions for sensors and Effective Sensor Sequences (ESS) of the nodes. The monitoring region represents the range that each sensor can monitor, while the ESS of nodes represents the collection of sensors that generate pressure responses when a burst occurs at each node, sorted by the magnitude of the response.

Next, in the Preliminary Regional Burst Localization Analysis (PRBLA) phase, by performing spatial intersection operations on the monitoring regions of multiple alarm sensors, the preliminary region of the burst is obtained. Finally, in the Precise Burst Localization Analysis (PBLA) phase, localization indicators are proposed to iterate through all the nodes for comparison to determine the precise location of the burst.

2.2. Effective Sensor Sequence (ESS) of Nodes

When burst events occur in a DMA, the monitoring pressure information provided by the sensors at different locations varies. This is influenced by factors such as the geographical distance between the burst event and the sensors, the connectivity of the pipeline topology, and other factors. Therefore, it is crucial to fully utilize the effective feedback information from the sensors for localization. In this study, the ESS of the nodes refers to the subset of sensors at a particular node that experience pressure fluctuations beyond a certain predefined range when a burst event occurs. This provides a data foundation for the next Precise Burst Localization Analysis (PBLA) stage in subsequent steps.

The proposed ESS in this method was derived based on the pressure sensitivity of the hydraulic model. The pressure sensitivity of a node involves traversing the various nodes in the WDNs. If there is a change in the water demand at a particular node, it causes a pressure change at that node, which, in turn, affects the pressure changes at other nodes [33,38]. Furthermore, the pressure sensitivity of a sensor involves traversing the various nodes and analyzing the effect of changes in burst amplitude at each node on the pressure changes at different sensors. The calculation process of the sensitivity matrix can be found in the literature [39,40,41]. The matrix was obtained through hydraulic model simulation analysis [33,38].

The sensitivity matrix $S_l^k$ and average sensitivity matrix $\overline{S_l^k}$ for node k with a burst flow rate of l is as follows:

$$S_l^k=\left\{\begin{array}{ccccc}\frac{r_{1,1}^k}{l}& \cdots & \frac{r_{i,1}^k}{l}& \cdots & \frac{r_{n,1}^k}{l}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ \frac{r_{1,j}^k}{l}& \cdots & \frac{r_{i,j}^k}{l}& \cdots & \frac{r_{n,j}^k}{l}\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ \frac{r_{1,t}^k}{l}& \cdots & \frac{r_{i,t}^k}{l}& \cdots & \frac{r_{n,t}^k}{l}\end{array}\right\}$$

(1)

$$\overline{s_n^k}={\displaystyle \frac{\sum _1^t\frac{r_{n,j}^k}{l}}{t}}$$

(2)

$$\overline{S_l^k}=\left[\overline{s_1^k},\overline{s_2^k},\dots ,\overline{s_n^k}\right]$$

(3)

where $r_{i,j}^k$ represents the difference between the normal and burst pressure data of sensor j at node k with the flow rate l at time t, $s_{n,t}$ represents the sensitivity value of sensor n at time t in the pressure matrix $S_l^k$, and $\overline{s_n^k}$ represents the average sensitivity values of sensor n at node k for timesteps 1 to t with a burst flow rate of l at node k, with the duration of each timestep being 15 min and the number of timesteps in a day being 96. Specifically, t is set to 96 in the study.

The calculation method involves utilizing the average sensitivity matrix $\overline{S_l^k}$ to construct the ESS of a node. Based on the matrix, the sensors with the highest sensitivity values correspond to the ESS of a node. In detail, the monitoring information of a sensor is directly correlated with its sensitivity. For a given node, a higher sensitivity value indicates a stronger perception capability on the part of the sensor. Consequently, sensors with higher sensitivity values are more likely to be included in ESS. Therefore, utilizing the sensitivity matrix from the sensors to determine the ESS of each node is highly effective.

The number of ESS is set to $n_{main}$. In $\overline{S_l^k}$, the top $n_{main}$ sensors with the largest sensitivity values are selected. The total sensor sequence of WDNs and ESS for node k is as follows:

$${Seq}^k=\left\{{List}_{index}\left[sorted\right(\overline{S_l^k}\left(i\right)]\right\}$$

(4)

$${Seq}_{main}^k=\left\{{Seq}^k[1,n_{main}]\right\}$$

(5)

where ${Seq}^k$ represents the total number of sensors of the WDNs sorted in descending order of sensitivity, the sorted function denotes the process of sorting the values inside the parentheses in descending order, ${List}_{index}$ represents a sequence formed by the indexes of the sensors corresponding to the sorted values, ${Seq}_{main}^k$ is the ESS of a node, which is reconstructed by extracting the first $n_{main}$ elements from ${Seq}^k$ from left to right, and n denotes the number of sensors, with $n_{main}$ ranging from 1 to n.

2.3. Deterimining the Monitoring Region of the Sensor

The entire water distribution network requires a network jointly monitored by multiple sensors, meaning that each sensor is characterized by the uniqueness of its monitoring region. The monitoring region of a sensor is the collection of indexes of burst nodes detected by that sensor [38,42]. It helps narrow down the area where the burst is located for regional burst localization. The construction method for the monitoring region of the sensor in this study was developed. When a burst event occurs at a node, based on the ESS ${Seq}^k$ of node k and the value of $n_{main}$, the monitoring sensor sequence of the node is determined. Specifically, in this study, the number of monitoring sensor sequences, $n_{mon}$, ranged from 1 to $n_{main}$. It can be observed that the monitoring sensor sequence of a node is a subsequence of ESS. The main method for determining the monitoring region of a sensor is that, if sensor i is in ${Seq}_{mon}^k$ of node k, then node k is included in the monitoring region of sensor i.

$${Seq}_{mon}^k={Seq}^k[1,n_{mon}]$$

(6)

$${Set}_i=\left[1,2,3,\dots \right]$$

(7)

where ${Seq}_{mon}^k$ represents the monitoring sensor sequence of node k, ${Seq}^k[1,n_{mon}]$ denotes the selection of the first $n_{mon}$ element from left to right in ${Seq}^k$, and ${Set}_i$ represents the monitoring region of a sensor obtained by ${Seq}_{mon}^k$.

2.4. Multi-Stage Burst Localization

In this study, based on the alarm sensors, the approximate preliminary region of a burst was quickly determined by Preliminary Regional Burst Localization Analysis (PRBLA), followed by further precise localization by Precise Burst Localization Analysis (PBLA). The two stages of burst localization analysis are discussed in Section 2.4.1 and Section 2.4.2. The following diagram illustrates the schematic representation of the two stages of burst localization (Figure 2).

2.4.1. Preliminary Regional Burst Localization Analysis (PRBLA)

When a burst event is confirmed within the DMA at a certain moment, the real-time pressure difference values should be calculated based on the monitoring data and sorted to obtain the sensors corresponding to larger pressure differences. These sensors are considered to be alarm sensors, and the number of these sensors is determined based on the scale of the WDNs. The intersection of the monitoring regions of the alarm sensors should be estimated to achieve PRBLA, which significantly reduces the localization area and improves the localization efficiency.

$${Set}_{PRBLA}={Set}_i\cap {Set}_j\dots {\cap Set}_w$$

(8)

where ${Set}_{PRBLA}$ represents the preliminary region of a burst and ${Set}_i$ represents the monitoring region of sensor i, while $i,j,w$ represent the node indexes of the alarm sensors.

In this study, the intersection of the monitoring regions of the alarm sensors was calculated to achieve PRBLA, reducing the localization range significantly and improving the efficiency of the localization process. Establishing a monitoring region for a single sensor to narrow down the burst location relies solely on that single sensor, whereas this study utilized multiple sensors to refine the burst location [42].

2.4.2. Precise Burst Localization Analysis (PBLA)

PBLA uses node analysis to identify and prioritize candidate burst nodes. This allows maintenance personnel to repair burst events based on their priority, significantly reducing the time required for burst maintenance. It facilitates early detection and prompt repair, thereby reducing losses and hazards caused by bursts.

(1)

Computation of dissimilarity-based localization indicator using a time window

In previous literature, some localization indicators have been used to measure simulated pressure residuals and measured pressure residuals. These localization indicators include Manhattan distance, Euclidean distance, Chebyshev distance coefficient, cosine similarity, Pearson’s correlation coefficient, Spearman’s rank correlation coefficient, and Kendall’s τ correlation coefficient [34,43,44]. Based on the ESS of each node, all nodes of the WDNs are traversed and the dissimilarity coefficient at time t is calculatedby comparing the real-time residual data of the unknown burst location with the model-based residual data. The dissimilarity coefficient d$(t,k,l)$ of the real-time monitoring data with burst flow rate l at node k at time t is defined as follows.

$$d\left(t,k,l\right)={\sum }_{i=1}^{n_{main}}{c}_i|{\displaystyle \frac{r_{i,t}^k}{r_{a_{i,t}}}}-1|,i\mathsf{∊}{Seq}_{main}^k$$

(9)

where $c_i$ represents the weighting coefficient for different sensors. A smaller value of d$(t,k,l)$ indicates a higher level of matching between the simulated burst event and the actual burst event, thereby increasing the likelihood of identifying it as a candidate burst location. In the ESS, the weighting coefficients for each sensor are determined based on the ranking. Taking ESS as four as an example, the weighting coefficients $c_1$ to $c_4$ correspond to the most effective sensor, the second most effective sensor, the third most effective sensor, and the fourth most effective sensor, respectively, with values such as 4, 3, 2, 1 or 2, 2, 1, 1.

To make comprehensive use of the data from multiple timesteps, the dissimilarity coefficient at time t was calculated as the average of the dissimilarity coefficients of a time window consisting of T consecutive timesteps. The calculation is as follows:

$$d\left(t_T,k,l\right)={\displaystyle \frac{1}{T}}{\sum }_{i=t_T+1-T}^{t_T}d\left(i,k,l\right)$$

(10)

$$d\left(t_T\right)={[d\left(t_T,\mathrm{1,1}\right),d\left(t_T,\mathrm{1,2}\right),\dots ,d\left(t_T,k,s\right)]}^T$$

(11)

where $d\left(t_T,k,s\right)$ represents the dissimilarity value of the time window at time t for node k and $d\left(t_T\right)$ represents the dissimilarity coefficient between each node and the burst flow rate at time t.

(2)

Localization analysis based on a dissimilarity localization indicator of multi-temporal and multi-sensor data

For each node, if it is a burst node, there exists a unique burst flow rate. Based on this principle, the actual burst flow rate for each node was filtered when a burst occurred. First, we calculated the dissimilarity of each node position under different burst flow rates and extracted the minimum dissimilarity value corresponding to the burst flow rate. If the node was a burst node, the burst flow rate was determined as the flow rate corresponding to the minimum dissimilarity value. Furthermore, within the preliminary region of the burst, we sorted the dissimilarities of each node and selected several smaller dissimilarity values. These selected burst positions were regarded as candidate burst locations, and the corresponding burst flow rates from the previous step were considered as the actual burst flow rates. A specific schematic diagram, as shown in Figure 3, illustrates this process. The preliminary region includes nodes k, w, and u, where each node has i burst flow rates. Based on the dissimilarity analysis, the burst flow rate corresponding to the minimum dissimilarity was extracted for each node. For node k, the smallest possible dissimilarity is $d\left(t_T,k,l_{k1}\right)$, corresponding to the burst flow rate $l_{k1}$. The dissimilarity of three nodes was then sorted to determine the burst location priority. If node k represents a burst, its burst flow rate is $l_{k1}$.

2.5. Evaluation Criteria

In this study, the effectiveness of the method was evaluated using a distance metric. The pipe distance was calculated as the shortest path from the candidate burst location to the actual burst location along the pipeline, such a method being closely related to the accuracy of burst localization [34]. The calculation formula for the pipe distance, which corresponds to the minimum path between the candidate burst node and the actual burst node, was obtained by retrieving the pipe lengths from the inp file using the EPANET toolbox.

$$D_{ij}={argmin}_{p_{ij}\left(k\right){∊P}_{ij}}{\sum }_{e_z∊p_{ij}\left(k\right)}{L}_z$$

(12)

where $D_{ij}$ represents the pipe distance corresponding to the shortest path between the candidate burst node i and the actual node j, $P_{ij}=\{p_{ij}\left(1\right),p_{ij}\left(2\right),p_{ij}\left(3\right),\dots p_{ij}(n\left)\right\}$ is the set of paths connecting node i and node j, $e_z$ represents the set of all pipes in the path, and $L_z$ represents the total length of the pipes in the path.

3. Results and Discussion

3.1. Basic Information

The proposed method has been successfully used in the C-Town network [45]. The specific information of the C-Town network is shown in Figure 4. The WDN is based on the Battle of the Water Network (BWN-II). BWN-II is an exercise that began in 1985 at the Water Distribution System Analysis Symposium, organized by ASCE Hydraulic Congress. Water is supplied to the system from a large reservoir (source) with a constant head reservoir (source) and seven balancing water tanks. The water is pumped through the pump station S1 to tanks T1 and T2, and the water supply to T2 is controlled by the TCV. Pumping stations S2 and S3 draw water to tanks T3 and T4, while stations S4 and S5 pump the supply from T1 to T5, T6, and T7. The following C-town WDNs were selected for this study. The selected WDNs consisted of 388 nodes, 429 pipes, one reservoir, seven tanks, and five pumping stations. The network topology was a combination of tree-like and loop-like structures. The D-town WDN consists of five existing DMAs. One DMA was selected as an example for the burst localization analysis. This DMA is primarily located near the pump station S1 and has a high pressure due to elevation imbalances, resulting in relatively large pressure differences among the nodes. In this study, this particular DMA was analyzed, including 132 nodes, 153 pipes, 10 pressure sensors, and a total demand of 118 L/s.

3.2. Data Preparation

EPANET 2.2 was used to perform the simulation analysis of the hydraulic data to facilitate the setting of working conditions under different external noises and bursts. The data simulation involved several steps [38].

(1)

Noise disturbance $r_d$ was added to the initial demand at each node to generate the 1-day water demand of the sensors.

(2)

The timepoint of burst occurrence $t_b$ and the duration $d_b$ were randomly set within a 24 h period, and the burst flow rate was randomly set within a given range. The sensor pressure of the burst events was generated.

(3)

Noise disturbance $r_p$ was added to the pressure data to generate the pressure data of the burst events.

(4)

The aforementioned steps were repeated to generate multiday pressure data.

In this context, $r_d$ represents the stochastic variation in daily water consumption by users due to factors such as weather, temperature, and other influences. The actual water consumption was determined by multiplying $r_d$ by the initial water consumption. $r_p$ denotes the measurement errors during the pressure acquisition process. This monitoring pressure was obtained by adding $r_p$ to the EPANET-simulated pressure. This is because $r_p$ can mask pressure drops caused by bursts, making detection more difficult [29]. In this article, both $r_d$ and $r_p$ follow a normal distribution [29].

In this study, five test sets with different burst flow rates were created to validate the effectiveness of the method. The fluctuation of water consumption $r_d$ and the measurement noise $r_p$ in the generation of the monitoring data were set to (0, 0.1) and (0, 0.2), respectively. The burst intensity ranges for the five test sets were: 3–9 L/s, 9–15 L/s, 15–21 L/s, 21–27 L/s, and 27–18 L/s. Each test set contained 130,000 days of data, with one burst event occurring per day, for a total of 130,000 burst events. In the model-based database, the simulated burst intensity range was 3–60 L/s, with a total of 58 discrete values. The total number of simulated burst scenarios was 7540 (130 × 58). For performance evaluation, the following additional parameter settings were used: the length of the time window T = 5, the number of effective sensors = 4, the parameters $c_1$ = 4, $c_2$ = 3, $c_3$ = $c_4$ = 1 were used in the calculation of the dissimilarity degree calculation during candidate burst node analysis.

The ESS of some nodes are displayed, as shown in

Table 1. The effective sensor sequence (ESS) of some nodes in DMA1.

Node Index	ESS (Sensor Index)	Node Index	ESS (Sensor Index)
148	354,167,202,245	27	6,277,245,202
145	354,167,202,245	6	6,277,245,202
146	354,167,202,245	3	6,277,245,202
166	354,167,202,245	342	6,277,245,202
152	354,167,202,245	342	6,277,245,202
251	167,202,148,245	71	245,277,202,162

. Specifically, the corresponding ESS for each of the nodes in the DMA when a burst occurred at those nodes. In this case, the number of effective sensor sequences (ESSs) $n_{main}$ was set to 4. Taking node 148 as an example, when the burst event occurred at that node, pressure changes, from largest to smallest, were detected at sensors 354, 167, 202, and 245 (node indexes). In other words, ${Seq}_{main}^{148}$ = [354, 167, 202, 245]. Additionally, it can be observed from Table 1 that different nodes may have the same ESS. For example, the ESS for nodes 148, 145, 146, 166, and 152 were the same. In detail, based on Table 1, when $n_{mon}$ = 1, the monitoring sensor sequence ${Seq}_{mon}^{251}$ of node 251 = [167], then ${Set}_{167}$ = [251], which means that the monitoring area of sensor 167 was node 251. When $n_{mon}$ = 2, ${Seq}_{mon}^{148}$ = [354, 167], ${Seq}_{mon}^{145}$ = [354, 167], ${Seq}_{mon}^{146}$ = [354, 167], ${Seq}_{mon}^{166}$ = [354, 167], ${Seq}_{mon}^{152}$ = [354, 167], ${Seq}_{mon}^{251}$ = [167, 202], then ${Set}_{167}$ = [148, 145, 146, 166, 152, 251], which means that the monitoring area of sensor 167 included nodes 148, 145, 146, 166, 152, and 251. As the value of $n_{mon}$ increased, the monitoring region range of the sensors expanded.

3.3. The Performance of the Multi-Stage Localization Method

3.3.1. The Performance of PRBLA

The probability in

Table 2. PRBLA results for different monitoring region sizes of the sensor.

Burst Flow Rate Range	${\mathit{n}}_{\mathit{m}\mathit{o}\mathit{n}}$ = 1 m = 1	${\mathit{n}}_{\mathit{m}\mathit{o}\mathit{n}}$ = 2 m = 1	${\mathit{n}}_{\mathit{m}\mathit{o}\mathit{n}}$ = 2 m = 2	${\mathit{n}}_{\mathit{m}\mathit{o}\mathit{n}}$ = 3 m = 1	${\mathit{n}}_{\mathit{m}\mathit{o}\mathit{n}}$ = 3 m = 2	${\mathit{n}}_{\mathit{m}\mathit{o}\mathit{n}}$ = 3 m = 3
3–9 L/s	27.69	39.23	21.54	60.00	26.92	24.62
9–15 L/s	73.92	86.92	36.15	86.92	40.76	25.38
15–21 L/s	76.15	86.92	40.00	94.92	56.92	26.92
21–27 L/s	85.46	88.46	44.61	96.76	57.69	30.76
27–33 L/s	90.85	93.76	50.00	98.08	60.77	31.00

determines the accuracy of the PRBLA. The higher the probability, the more accurate the PRBLA. By statistically calculating the probability of the actual burst being in the preliminary area of the burst, the effect of the current stage of PRBLA can be evaluated. If the number of alarm sensors m is too large, the intersection of the monitoring areas of the alarm sensors may not result in a common area.

As it can be seen from Table 2, when m (the number of alarm sensors) = 1 and $n_{mon}$ (the number of monitoring sensor sequences) ranged from 1 to 3, and when the burst flow was between 27 L/s and 33 L/s, the probability that the current burst node was in PRBL was more than 90%. When m = 2, the probability of the actual burst being in the initial region decreased significantly. When $n_{mon}$ = 2, the probability was 50%, and when $n_{mon}$ = 3, the probability was 60.77%. As m increased, the number of alarm sensors used for the intersection increased and the range became smaller, leading to worse regional location effects in PRBLA.

Table 3. The preliminary region does not include the true burst location or the probability of any node.

Burst Flow Rate Range	${\mathit{n}}_{\mathit{m}\mathit{o}\mathit{n}}$ = 2 m = 2 (Excluding the Burst)	${\mathit{n}}_{\mathit{m}\mathit{o}\mathit{n}}$ = 2 m = 2 (without Any Nodes)	${\mathit{n}}_{\mathit{m}\mathit{o}\mathit{n}}$ = 3 m = 2 (Excluding the Burst)	${\mathit{n}}_{\mathit{m}\mathit{o}\mathit{n}}$ = 3 m = 2 (without Any Nodes)	${\mathit{n}}_{\mathit{m}\mathit{o}\mathit{n}}$ = 3 m = 3 (Excluding the Burst)	${\mathit{n}}_{\mathit{m}\mathit{o}\mathit{n}}$ = 3 m = 3 (without Any Nodes)
3–9 L/s	78.46	60.00	73.08	31.53	75.38	43.07
9–15 L/s	63.85	51.53	59.24	34.61	74.62	71.53
15–21 L/s	60.00	37.69	43.08	33.84	73.08	71.53
21–27 L/s	55.39	37.69	42.31	33.84	69.24	67.23
27–33 L/s	50.00	30.00	39.23	35.32	69.00	55.23

shows the probability that the initial burst region did not contain the actual burst and the probability that the intersection of the sensor monitoring areas did not contain any nodes when m = 2 and m = 3. Specifically, when $n_{mon}$ = 2 and m = 2, the probability that the region did not contain the actual burst was 78.46%, and the probability that it did not contain any nodes was 60%. When m > 1, the main reason for the poor localization result was that the number of alarm sensors was m. When the monitoring areas of these m sensors were intersected, in most cases, the result was an empty set that did not contain any nodes. This is because the differences in the monitoring area of the sensors in this WDN are quite large, with some sensors differing by more than 40 m. When a burst occurs at most nodes, although it may cause pressure fluctuations in $n_{main}$ sensors, the pressure fluctuation values of the remaining sensors are similar. When subjected to noise interference, the sensor sensitivity ranking based on the model may not be consistent, and, if the m value in DMAs is large, it will affect the regional location effect. At the same time, it is also related to the layout and number of sensors. If only one sensor experiences a relatively large change, while the others also change but not significantly, the intersection and sorting during the localization process may be inconsistent with the sensor monitoring area obtained based on the model.

3.3.2. The Performance of PBLA

Figure 5 shows the topological distance between the optimal candidate burst node and the actual burst node for different ranges of burst flow rate and preliminary burst region. When the burst size was in the range of 3–9 L/s, the average number of nodes included in the preliminary region ranged from 23 to 32 or 34 when $n_{mon}$ varied from 1 to 3. This accounts for approximately 17.69%, 26.15%, and 24.61% of the total number of nodes in DMA1, respectively. In other words, compared with traditional model-based residual-based localization methods, the PBLA phase narrowed down the burst region by 82.31%, 73.85%, and 75.39%, respectively. For the burst flow rate of 9–15 L/s, the average number of nodes in the preliminary region ranged from 24 to 40 as $n_{mon}$ varied from 1 to 3. This corresponds to approximately 18.46%, 25.38%, and 30.76% of the total number of nodes in DMA1, respectively. The PBLA phase narrowed down the burst region by 81.54%, 74.62%, and 69.24%, respectively.

When the burst flow rate was in the range of 21–27 L/s, the average number of nodes in the preliminary region ranged from 24 to 38 when $n_{mon}$ varied from 1 to 3. The PBLA phase narrowed down the burst region by 81.54%, 75.39%, and 70.77%, respectively.

In the case of the burst flow rate being in the range of 27–30 L/s, the average number of nodes in the initial region ranged from 25 to 38 when $n_{mon}$ varied from 1 to 3. The PBLA phase narrowed down the burst region by 80.77%, 74.62%, and 70.77%, respectively. From Figure 5, it can be observed that the final localization results for different burst magnitudes with varying values of $n_{mon}$ and m = 1 were not significantly different. In the PBLA phase, the probability of the actual burst node falling within this region was similar and attained more than 90%. In the second phase of localization, accurate and precise burst localization can be achieved. Overall, the results show that, regardless of the burst magnitude and the number of monitoring nodes $n_{mon}$, the localization performance is consistently effective and accurate.

According to the results shown in Figure 5, focusing on the analysis with m = 1 and $n_{mon}$ = 2, the cumulative probabilities for the optimal candidate nodes within a distance of 500 m in the four burst flow rate test sets were approximately 32.30%, 61.53%, 64.61%, and 66.15%, respectively. In this context, the average length of each pipe segment in the DMA was approximately 200 m, and “within 500 m” referred to the localization result being either at the actual node or within the nearest one-to-two nodes. The cumulative probabilities for the distance values within 800 m were approximately 37.69%, 75.38%, 79.23%, and 80.76%, respectively. Here, within 800 m refers to the localization result being either at the actual node or within the nearest one-to-four nodes.

The localization results with distance values indicating relatively distant pipes could be affected by external noise, which could cause variations in sensor pressure fluctuations induced by the burst to be interfered with. Additionally, nodes located at the end of the pipeline may not be within the monitoring range of the sensors, resulting in a lower impact on sensor pressure fluctuations caused by the burst events. Overall, the analysis shows that the cumulative probabilities decrease as the distance from the optimal candidate node increases. This suggests that the localization accuracy decreases for pipes farther away from the candidate nodes, potentially due to external noise interference and the limited monitoring range of sensors at the network’s end.

Figure 6 shows the burst localization accuracy for burst events under different flow rate ranges. The cumulative probability curves for node_1st, node_2nd, and node_3rd show a consistent trend, significantly outperforming node_18th, node_19th, and node_3rd. The analysis of localization indicators plays an important role in the accurate localization process by effectively quantifying the probability of the actual candidate burst nodes. The burst node identification team should prioritize detection based on node_1st, node_2nd, and node_3rd. The search direction should focus on nodes with the highest probabilities, directing attention to the adjacent nodes with the next highest detection priority. Overall, the analysis highlights the significance of localization indicators in accurately identifying burst nodes and directing the search process to nodes with the highest probabilities.

4. Conclusions

This study proposes a novel multi-stage burst localization method for DMAs based on spatio-temporal information. This study performs Preliminary Regional Burst Localization Analysis (PRBLA) and Precise Burst Localization Analysis (PBLA) by exploiting the dissimilarity and correlation among data from multiple timesteps and multiple sensors. In four different burst magnitude test datasets, the cumulative probabilities for the optimal candidate nodes within a 500 m range were approximately 32.30%, 61.53%, 64.61%, and 66.15%. This figure corresponds to the probability that the localization result corresponds to the actual node location or falls within the one-to-two closest nodes in its topology structure. This approach improves the accuracy and efficiency of burst localization and provides valuable guidance for actual burst localization. The main findings based on the case study are as follows:

• The pressure fluctuation patterns recorded at sensors triggered by burst events are completely different from those recorded under normal conditions. Based on this, the method proposed in this study utilizes key spatio-temporal information features using hydraulic models to suggest the effective sensor sequences of nodes and the monitoring regions of sensors. By using different sensor monitoring data resulting from different burst events, this study overcomes the low accuracy associated with the use of single pressure gauges in the literature.
• The effectiveness of the method is related to the settings of the monitoring region of the sensors and the number of alarm sensors. In cases where the monitoring region of the sensors is small, and the alarm sensors are too numerous, multiple sensors sharing a small or even empty monitoring region will lead to poor localization results. Under reasonable combinations of these two parameters, the proposed method effectively locates burst events, providing a viable solution for burst localization.
• The method performs a two-stage progressive localization process, gradually narrowing down the space until precise localization is achieved. In the Preliminary Regional Burst Localization Analysis (PRBLA), the localization range is significantly reduced. In the Precise Burst Localization Analysis (PBLA), the proposed dissimilarity-based localization indicator can accurately locate specific nodes and identify multiple nodes prioritized by localization accuracy. This method addresses the problem of low efficiency in the localization of large practical networks and holds promising prospects and guiding significance for practical engineering.

The method of this study relies heavily on the hydraulic model, and its performance is affected by the availability and accuracy of the model. Uncertainties may include modeling errors, node demand uncertainties, and measurement noise. In addition, the computational costs and uncertainties in parameter estimation can also limit the applicability of the model-based methods. In future research, the dependence on the hydraulic model will be appropriately reduced, and knowledge from deep learning and graph theory will be combined with other burst localization methods to further improve the accuracy of burst localization.