Pressure gauge network design for leakage detection in water distribution systems using a combination of the pressure residual vector method and the entropy concept

Water distribution systems (WDSs) play a crucial role in providing clean water and making it accessible to the population. However, the problem of leakage, along with other issues reported by many researchers (Mays 2000; Kanakoudis & Tolikas 2001; Kanakoudis 2004a, 2004b), in the distribution network is a persistent challenge faced by water supply companies worldwide. Leakages affect the system's efficiency, reduce the available water supply, and increase costs for repairs and maintenance. In addition, they also pose a serious threat to the quality of water provided to consumers, as the infiltration of contaminants into the network can lead to the transmission of waterborne diseases. Therefore, addressing the problem of leakage in the water distribution network is of utmost importance. Effective leak detection and repair mechanisms and the regular maintenance of the system can help water supply companies to minimize the impact of leakages on the water supply, conserve precious resources, and maintain the quality of the water being supplied.

Various techniques have been employed to detect leaks, which can be categorized into direct and indirect methods (Puust et al. 2010; Li et al. 2015; Adedeji et al. 2017; Al-Washali et al. 2020; Hu et al. 2021). Direct methods involve using observational tools to detect leaks within a specific area, while indirect methods require measuring and analyzing signals to identify potential leaks. Due to the lack of continuous monitoring capabilities in direct methods, indirect methods have gained wider use and advancement (Hu et al. 2021). Among the indirect methods, those that study the effect of leakage on pressure variations in different areas of the pipelines or systems are the most convenient and widely used (Pérez et al. 2010, 2011; Casillas et al. 2013; Adedeji et al. 2017; Salguero et al. 2018).

Analytical methods such as the hydrostatic pressure testing method (Edwards 2014), inverse transient analysis method (Wang et al. 2002; Haghighi & Ramos 2012; Sousa et al. 2014), and pressure-flow deviation method (Fukushima et al. 2000; Zhang 2001) are suitable for pipelines, while processes based on the continuous monitoring of pressure variations are useful in detecting leaks in WDSs. The pressure residual vector (PRV) method is one such process that is based on the continuous monitoring of pressure variations (Ponce et al. 2012; Casillas et al. 2013; Abdulshaheed et al. 2018). The PRV method measures the pressure drop in pipe sections, constructs a hydraulic model based on pressure readings collected from sensors installed on the system, and calculates a residual vector. Analyzing the resultant vector can identify potential leaks. Effective leak detection requires the proper analysis and interpretation of observed results and compares them to reference data derived from hydraulic or numerical models. Therefore, a network of pressure monitoring sensors, reference data, and instructions on interpreting and comparing the data are the three primary elements for leak detection and location using the PRV method.

A quantitative indicator known as entropy content has been suggested by several researchers (Tanyimboh & Templeman 2000; Setiadi et al. 2005; Tanyimboh et al. 2011) as a general performance indicator for WDS. This concept is commonly used as an objective function in designing different parts of a WDS (Tanyimboh & Sheahan 2002; Perelman & Ostfeld 2007; Saleh & Tanyimboh 2014). Recently, the entropy content has also been utilized to express the pressure distribution in WDS, which facilitates operational actions such as leakage and failure management, backwater intrusion, and demand control (Christodoulou et al. 2013; Ridolfi et al. 2014; Khorshidi et al. 2020). The performance of a WDS can be indicated by the information content obtained from the calculated and measured pressure difference. However, these studies measure entropy content based on observed data for a specific period, making it challenging to apply them in practice and in different conditions, such as varied water consumption during the day, night, summer, and winter.

This study is part of research aimed at detecting and locating leaks in a WDS. The innovation in this research involves incorporating the entropy content in the PRV method and the use of the probability distribution of water consumption to calculate the entropy content. This approach allows for a consideration of a probability structure in water consumption, which is lacking in previous studies. The main objective of the study is to identify the nodes in the WDS that are most likely to be affected by leakage events in other parts of the system (pipes). The algorithm involves assigning the hourly water consumption pattern to the design model of the pressure gauge network (PGN) and calculating the entropy content of pressure variations in the normal state (without leakage) and with leakage. Nodes that are most likely to be independently affected by leakage in different parts of the WDS and have the most information are identified as potential locations for PGN installation. The location of the leak will be determined through parallel research using information obtained from the strain pressure at the selected nodes.

The scope of the current study encompasses several topics that are discussed in this section. The measurement of entropy content serves as the computational core of this research. Additionally, the section explores an alternative approach to analyzing water consumption utilizing probability distributions rather than the original dataset. To achieve this, a series of probability density functions (PDF) based on a one-year-period observation are presented. The section also introduces hydraulic computational relations that are utilized in the PRV method. Finally, a decision-making algorithm is presented that summarizes the strategy for selecting nodes for pressure gauge installation.

Water consumption is a highly complex process that is influenced by several factors. Climate conditions (Lee & Dang 2019), seasonal variations in temperature and water availability (Machingambi & Manzungu 2003; Arouna & Dabbert 2010), water supply limitations (Andey & Kelkar 2009), tariff systems and pricing (Renwick & Green 2000), household characteristics (Syme et al. 2004; Shove et al. 2010), and attitudes toward water conservation (Corral-Verdugo et al. 2002) are among the factors that can directly or indirectly impact water consumption and usage behavior (Jorgensen et al. 2009; Kanakoudis & Gonelas 2014). To manage water distribution networks, typical patterns for specific climate conditions and social categories are usually employed, although the efficiency of these models can be improved by gathering field data.

The study utilized 39 electromechanical water meters with automatic meter reading (AMR) capability to collect semi-real-time data from single-family terraced homes in Koohsabz village. The village, located in southwest Iran (30.0°N, 52.7°E), has an average population of 1,300. The study was conducted between September 23, 2018, and September 23, 2019, during which water consumption data and other related information were gathered. The region spans an area of 0.32 km² and has a semi-arid climate. Water supplied by the county water department is only used for sanitary and houseplant irrigation purposes in the studied homes according to the Statistical Center of Iran (2015) reports. To evaluate hourly per-capita water consumption, the head of the household completed an electronic form that required information about the daily number of inhabitants in the house and their times of entry and exit.

The electromechanical water meters combine electronic circuit components with mechanical water meters to provide automatic functionalities. The measurement basis of the system is mechanical, where water consumption is tracked using an impeller meter. The meter readings are then digitized through a transducer and stored on a micro-SD memory card every hour. In residential settings, these devices are programmed to automatically send gathered readings as text messages through the Global System for Mobile Communications network to a receiver modem once every 24 h. These text messages are subsequently decoded on a PC connected to the modem and loaded into a database for further processing.

WDSs undergo evaluation from different perspectives, including reliability, cost, health risks, and pressure management. Notably, the current research process views information content retrieved from the system as an evaluation metric, considering possible accidents like leakages. This paper presents three subsections which expound on Shannon's entropy and joint entropy's fundamental theory, the approach used in designing PGNs, and the hydraulic analysis model incorporated in the study.

To analyze a pipe network, which includes point leakage, Equations (10) and (12) can be combined to create a system of equations. This system also incorporates the leaking point as a node. By solving this system of equations, the pressure head among nodes and the quantity of leakage can be determined. However, Equation (9) causes the system of equations to be nonlinear. To achieve accurate solutions, an iterative process is necessary to solve the equations.

In this study, the design of a PGN involves a combinatorial search, which aims to identify the best configuration of pressure gauge placements among all available nodes to achieve full coverage. To explore the entire search space, options must be evaluated to identify the optimal solution. However, due to the high number of options to be considered and the significant computational effort required for each analysis, a comprehensive coverage of the search space is a time-consuming process.

Increasing the number of pressure gauges in the WDS can lead to a rise in joint entropy, which is a measure of the information obtained from the system. However, a higher number of gauges may not necessarily ensure a significant increase in the retrieved information. Therefore, the first step is to identify the pressure gauge with the highest entropy value. Subsequently, the dual composition of the gauges is examined to identify the pair with the maximum entropy value. The number of gauges is then increased, and the process is repeated until further additions fail to significantly increase the joint entropy. At this point, the search process is terminated.

The following proposed algorithm summarizes the stages for PGN design to detect leakage in a WDS:

Detecting leaks in a WDS can be achieved by monitoring pressure variations through the use of a PGN installed in the system. It is important to note that pressure variations caused by factors other than leaks, such as changes in consumption, contribute to the total pressure variations in each node. Therefore, to detect leaks, it is essential to differentiate pressure variations caused by leakage from those caused by other factors (Shirzad et al. 2013). In addition to this, selecting the appropriate index is crucial in comprehensively reflecting the information content of pressure changes. Entropy is a reliable indicator of pressure variations that can significantly reflect the information content. This paper discusses the effects of leakage on the entropy of pressure variations at different nodes for two WDS.

In the presented study, the research process consisted of analyzing and discussing the results of two WDS implemented with different pipe configurations. The primary objective was to assess the effect of pipe leakage on the entropy of pressure variations at every node. To determine this, numerous leakage parameter values were examined as per Equation (12), and the investigation aimed to understand how these parameters influenced the overall information content.

In simulations, leakage can be modeled throughout the entire pipe, at a particular point in the pipe, or at fittings. In this study, a leak is hypothetically created at a mid-point in the pipe to impact two nodes located on either side of the pipe. Equation (12) serves as the governing equation for leakages with values of c and in the range of 0–0.8 m³/s and 0.5–1.5, respectively. A value of represents no leakage, while higher values of emitter coefficient (⁠⁠) indicate a greater leakage rate. Meanwhile, denotes the effect of pressure on the leakage rate at the site of leakage, which varies depending on the material used in the pipe (Wu et al. 2010).

In addition to variations of entropy values with leakage parameters, the combination of nodes with maximum joint entropy can be selected as the preferred nodes for the installation of pressure gauges to monitor leakage in the WDS. Due to the possibility of different results with different hydraulic leakage parameters, Table 2 shows the rank-order analysis for Examples 1 and 2 with different leakage parameters (⁠ and ⁠). In this table, first, the leak parameters and the number of assumed gauges, the selected nodes, and the percentage of occurrence of these options for each rank are shown. For instance, in Example 1, for and ⁠, if the goal is to find three nodes with maximum joint entropy, in 56.1% of cases, nodes 2, 3, and 6 are selected. Then, it can be seen that these results are relatively repetitive for all leakage parameters. In other words, in this example, the PGN design for leak detection in the WDS is independent of the leak parameters. A similar result can be deduced in Example 2, representing a regular pipe-configuration WDS, and as a result, in this example, it can be also concluded that the PGN design does not depend on leakage parameters.

Table 2

Rank-order analysis for selecting nodes with maximum entropy and joint entropy for the PNG design

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Determining the number of optimal pressure gauges and selecting them in the next step pave the way for designing a sensible PGN in Examples 1 and 2. In Example 1, if nodes 2, 3, and 6 are selected for the installation of leak detection pressure gauges (from the optimal number and nodes proposed in Figure 9 and Table 2, respectively) in different leak parameters, about 90% of the joint entropy of water head variations due to possible leakage in the pipes can be absorbed. Moreover, in Example 2, with reference to Figure 10, the number of optimum pressure gauges in the PGN for leak detection is considered to be four. However, as noted in Table 2, selecting the optimal number of pressure gauges for this example is not as clear as selecting the s pressure gauges selected in Example 1. However, the proposed pressure gauges for leak parameters in the critical interval can be more reasonable, and in this example, nodes 5, 15, 21, and 23 are suitable candidates for installing pressure gauges. However, it should be noted that this number of pressure gauges does not saturate the content of joint entropy obtained from the whole system, and only the rate of growth of this quantity decreases with increasing the number of pressure gauges.

It is observed that the entropy content calculated in a small period of time yields a lower percentage of information, and this information is not sufficient to locate the leakage properly. However, a long time, despite increasing the accuracy of leak detection, causes more costs and reduces the efficiency of the method. As can be seen in this figure, according to the data observed in the two nodes (representing the other nodes of the water distribution networks in Examples 1 and 2), about 40 and 80 h of monitoring of water head variations is required to achieve, 70 and 80% of the total entropy, respectively. To reach more accuracy, it is necessary to increase the monitoring time to 360 h (15 days) to obtain 90% of entropy; however, this is not defensible and reduces the efficiency of the method. Therefore, in these two examples, it is suggested that entropy calculations for leak detection are based on a time of about 60 h that can represent three-quarters of the total entropy.

WDSs commonly face the challenge of leakage, which has been the focus of many studies over the past three decades. Researchers have explored various aspects of leakage management, including leak identification and control. One particularly interesting area of research is the detection of leaks, for which the PRV method has been widely used due to its cost-effectiveness and practicality. Research has concluded that water head variations in nodes do not always have a proportional effect on leakage (Zyl & Clayton 2007; De Marchis & Milici 2019). This is because small amounts of leakage can impact a large area of the system due to the resonance phenomenon. The study confirms these findings and demonstrates that even for small leakages, the average entropy content obtained from nodal water head variations is significant. For example, in both WDS examined in this research, an emitter coefficient of resulted in a substantial increase in mean entropy content from around 0.2 to over 6. Therefore, this proposed method can effectively detect even small leaks.

Few previous studies have employed the idea of entropy in the PRV method for leak detection (Geng et al. 2019; Khorshidi et al. 2020). However, both this and other models of leak detection suffer from two significant drawbacks that the current research aims to address. The first issue concerns the dataset used to create the model. Methods that combine entropy and the PRV method rely on a particular dataset. To overcome this limitation, the present study uses a probability distribution model based on water consumption patterns. Calculations are performed by considering a realization of the probability distribution function, which leads to a more robust and comprehensive approach. The second issue pertains to the monitoring intervals for PRV-based leak detection methods, which have received comparatively less attention in the existing research. While some studies suggest taking pressure measurements during specific times of the day – such as in the morning when values are typically highest (Wu et al. 2010) – or over extended periods (Abdulshaheed et al. 2017), the impact of monitoring time on the precision of leak detection has not been extensively investigated. In this study, we assessed the efficiency of PGNs across various monitoring time intervals by analyzing the rate of increase in entropy content over time.

Despite the positive aspects of this research, it is important to acknowledge that there are other ideas that could potentially improve the performance of the presented model. For instance, when expanding the WDS, it is no longer feasible to search for all possible combinations for PGN design. Therefore, an optimization method such as genetic algorithms or simulated annealing should be utilized to complement this model. Furthermore, the governing model of this method, like many other existing models, is based on steady equations. Therefore, exploring this method using unsteady hydraulic equations could be a topic of future research to enhance the stability of this model.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.