AbstractThe presence of leaks in water distribution networks (WDN) is a problem of global concern that causes significant annual losses of treated water. Most efforts to solve this problem concentrate on the development of methods to help locate leaks in the least possible time. However, the decision to repair leaks is not always trivial because of their uncertain location, size, and other related consequences. For example, intervening places of difficult access or those that would generate major city disruptions may trigger the decision of not taking any action. This paper presents a methodology to determine the optimal location of pressure sensors for leak detection considering the concept of value of information (VOI) using a model-based optimization framework. The methodology allows for estimating the potential value of the data provided by a sensor in terms of its usefulness to assist decisions. The methodology, which also describes how to interpret VOI results, was applied to one of the water districts of a city in Western Europe. Results showed that the optimal VOI-based sensor placement outperformed the existing pressure sensor network for leak detection.IntroductionWater resources are under unprecedented pressure due to global issues such as population growth, rapid urbanization, environmental pollution, socioeconomic development, and climate change (WWAP 2017). One important action to mitigate this pressure is to be more efficient in the use of available resources. In water distribution networks (WDNs), this implies performing actions including water demand management and conservation programs aimed to reduce losses of treated water during transport, caused, for example, by leaks or unauthorized connections. Although some of these actions have reduced water losses in the last decades, a significant volume of water is still lost in many systems. According to Liemberger and Wyatt (2018), the global annual volume of water lost was estimated as 126  billion m3, equivalent to 75% of West Africa’s Niger River or half of India’s Ganges River disappearing, with the potential to supply 800 million people if these losses are reduced by a third (Dalton 2018). Financially, this is worth USD 39 billion per year, equivalent to the GDP of Cameroon in 2020.Apart from financial losses, leaks may also trigger other problems, such as the possible water quality deterioration due to the intrusion of substances into the system (Gong et al. 2016; Farley et al. 2009). Small but continuous leaks can wash away soil, eroding important infrastructure (Ali and ho Choi 2021), and cause the uneven distribution of pressure and flow in the network, generating loss of trust by users toward water entities (Quintiliani et al. 2019).Consequently, water utilities regularly work on applying methods that help locate leaks in the least possible time. Some of these methodologies are based on the exact location and physical visualization of the leak (Farley and Trow 2005). However, although leaks could be pinpointed directly when they are visible, these approaches are not always appropriate for large systems because of economic limitations (Sanz et al. 2016). Hence, a number of methods have been developed to alert about the possible existence of leaks and to provide information about possible areas where the source of water loss may be originated. These methods depend mainly on secondary evidence of leaks, such as checking unexpected changes in the network’s pressure values (Raei et al. 2019). Indeed, the presence of leaks may increase the flow and velocities in the network, causing larger hydraulic losses, leading to a readjustment of pressures in the nearby pipes.A common way to estimate the expected pressure values in a network is by using computer models, which are mathematical representations of the entire network that could simulate their behavior for a range of functioning scenarios. Furthermore, real pressure values are obtained from measurements in the field with the help of pressure sensors. The difference between expected and measured pressure values can be an indication of the potential existence of a leak (Raei et al. 2018).Although installing a large number of sensors in a network could be ideal to detect leaks, this is not always possible. In some cases, due to lack of budget or technical difficulties, water utilities only have access to a limited number of sensors, making it challenging to capture pressure drops in the whole WDN. However, a limited number of sensors could help identify the presence of leaks in most parts of the network if they are optimally located, potentially reducing reparation time, loss of water, and associated damages (Bohorquez et al. 2020).Multiple investigations have been developed to build computational frameworks for the optimal location of sensors. For example Farley et al. (2009), Pérez et al. (2009), Sanz et al. (2016), Raei et al. (2018), and Quintiliani et al. (2020) presented alternatives for sensor deployment using the model-based fault diagnosis theory, where the pressures of a system assumed to have no leaks are compared with those under leak conditions. Their results showed that the performance of the sensors located using computational optimization frameworks outperformed the performance of sensors located following empirical practices. Additionally, it was concluded that there is an optimum number of sensors for the identification of leaks, and that the addition of more sensors improves marginally the quality of this identification.Steffelbauer and Fuchs-Hanusch (2016), Jung and Kim (2017), and Taravatrooy et al. (2020) incorporated uncertainty analysis to discuss the effects of leak detection accuracy due to errors in measurements from pressure sensors and demand variation, concluding that points sensitive to leaks may also be points sensitive to uncertain in demand and pressure variations and, therefore, are less suitable for sensor placement. Blesa et al. (2014) found that, in their method, the optimized sensor locations were not sensitive to the size of the leaks, and Raei et al. (2019) found that the variation of a threshold for which the pressure drop start to be significant may also have a negligible effect on sensor locations. Recently, the use of information theory, a framework to quantify information content, flow, and losses in communication systems (Shannon 1948), has been used to assist monitoring in water systems (Alfonso 2010) and has been applied by Khorshidi et al. (2020) to quantify the ability of sensors to detect leaks based on the uniqueness of the information they provide.The problem of leak repairs in WDN can be formulated from the decision-making-under-uncertainty perspective. Apart from visible leaks, which can easily be repaired if the pipes are accessible, the decision to repair suspected leaks at places that are not accessible is a difficult one. For example, immediate intervention can be very disruptive and costly (e.g., in highways), and this situation can be aggravated if the initial information that triggers the intervention is not reliable. However, if no intervention is carried out, the volume of lost water can be financially significant.Pipes prone to leaks may be identified based on historical records of previous events (e.g., reparations, complaints, and surveys), characteristics of the network (e.g., pipe’s material and age), and external aspects (e.g., traffic load, tree roots in the vicinity, and so on) (Christodoulou et al. 2009). However, other aspects involving deficient operation (e.g., water hammer hits, pump scheduling, and pipe and joint clogging), as well as unknown construction aspects (e.g., deficient founding, incorrect installation, poorly constructed joints, and so on) bring uncertainty to the problem and may influence the perception about the condition of the network. This uncertainty, in turn, can be reduced by investing in extra information, for example, by setting up a network of sensors to report pressure variations, which can be used to estimate the presence and location of leaks. The new information, however, may not be perfect, producing wrong indications about the state of the system and possibly triggering wrong decisions.This paper addresses this particular aspect with the concept of value of information (VOI), a concept developed in the 1960s with the work realized by Grayson (1960) and conceptualized in the work of Hirshleifer and Riley (1979). Since then, VOI has been used in many sectors and disciplines, including the oil and gas industry (Holmes 2021). In terms of monitoring network design, VOI has been used to detect and reduce flood impacts (Alfonso and Price 2012; Alfonso et al. 2016), assess water quality (Shaqadan 2008), and design groundwater level monitoring networks (Khorshidi et al. 2018). VOI has started to be recognized as a valuable tool for decision making under uncertainty in engineering (Zou et al. 2021). Recently, a first attempt to apply VOI for the optimal location of sensors, using a demand-driven analysis (DDA) has been proposed by Khorshidi et al. (2020), assuming that each node can be assigned a VOI, and those with the maximum value could be selected as the best candidates to locate a sensor. However, their work, which is largely based on the ideas of Alfonso and Price (2012), had an unclear interpretation of the VOI parameters, and as a consequence, the perception of the decision maker seemed to be ignored.Although important progress has been accomplished in the development of methodologies for optimal localization of pressure sensors, there are still gaps that need to be addressed. First, most approaches rely on DDA simulations, which assume that all demands are supplied regardless of the network’s pressure values. This assumption is distant from reality because a reduction in pressure may cause that demands are not supplied, an aspect that is better addressed using pressure-driven analysis (PDA) (Braun et al. 2017). Second, current methodologies give exclusive attention to quantitative hydraulic parameters such as detection time, water loss, number of sensors, and number of leaks detected, among others; they ignore qualitative factors such as expert knowledge about the WDN, which could define criteria to give higher priority to leak detection in areas with a greater impact or with a higher tendency for the presence of leaks. In these situations, experience is extremely valuable yet largely ignored in the existing methods to optimally locate sensors.This paper proposes a methodology to fill these gaps, first, by considering PDA approach to simulate leaks to optimize the location of pressure sensors, and second, by representing the decision maker’s judgments using parametric representations based on the VOI concept.Value of Information: Background and EstimationDaily decisions are normally based on preconceived concepts or criteria derived from experience; for example, a cloudy sky or high humidity (states s of the system) may increase people’s belief about the occurrence of an imminent rainfall event and may influence the decision to take an umbrella or going out without it (the actions a available to the decision maker). This initial belief, however, could be influenced in the light of new information such as the weather forecast (which generates messages m about the possible state of the system). Depending on the accuracy of the message and the previous experiences of the user with it, the initial decision of taking the umbrella or not may change. The extent to which the decision changes with respect to new information can be investigated with the VOI concept.This approach, which is based on Bayesian updates, is based on three parameters, namely the prior belief that the decision maker has about the state of a system before the acquisition of new information (πs), the consequences or costs of taking an action in relation to each possible state of the system (Cas), and the quality of new information to be received, described by the conditional probability of receiving a message given the state of the system (qm,s). If the decision maker is convinced that one of the states is happening in the system, then a probability of 1 is assigned to such state and a probability of 0 is assigned to the remaining states. In this extreme case, any extra information has no value for the decision maker to decide. On the contrary, if the prior beliefs are equally distributed among the available states, the decision maker is said to be ignorant and any additional information has value for decision making.In this paper, the word utility refers to the benefit of performing an action, given the consequences of such action, and it can be estimated using the Neumann-Morgenstern expected utility rule, defined as in Eq. (1) (1) where S = set of the possible states of the system. The decision maker should select the action a0 that gives the highest utility, therefore using Eq. (2) (2) Once the new information is acquired, the decision maker can either rely on this new information or ignore it, the case in which the final decision will be based only on the prior belief. The new information, represented in the message m, could influence the perception of the initial state of the system. Accepting the new message implies that the prior belief is updated; in the Bayesian framework, this update can be estimated as in Eq. (3) (3) πs,m=qm,s×πsΣsqm,s×πswhere πs,m = updated perception of the system states given the message m; and qm,s = probability of receiving a message m in each of the system’s states. Once the prior belief is updated and the posterior belief is obtained, the utility of taking actions using this new information is evaluated as Eq. (4) (4) u(a,πs,m)=∑sCa,s×πs,mTo evaluate the value of the messages, the difference between the utility of performing an action based on prior beliefs and the utility of acting once new information is acquired is calculated by Eq. (5) (5) Δm=max{u(a,πs,m)}−max{u(a,πs)}The total VOI is the summation of products of the value of each message and the joint probability of receiving a message in a given state of the system. Hence, VOI can be estimated as in Eq. (6) (6) VOI=∑m(∑sqm,s×πs)×ΔmMethodologyThe proposed methodology applies the VOI concepts to find optimal locations for pressure sensors, and it is described using the following notation: i refers to a node that is leaking, simulated as a node with extra demand; j refers to any node in the network that is considered as a candidate sensor location; J is a set of n sensors, J={j1,j2,…,jn}, located in different candidate nodes. The methodology is structured in three main parts. The first part consists of assessing the capacity of each node in the network to capture pressure disturbances caused by leaks. To this end, an analysis of the network in terms of hydraulic parameters is required, with the generation of pressure deviation matrices. The second part of the methodology is related to the application of the VOI concept, using it as an objective function to select the optimal set of sensors. To this end, states, actions, and messages are defined as follows: the state s of a node can be either s1, Leak, or s2, No leak; the corresponding available actions (a) to take are a1, Intervention, and a2, No intervention. The messages m to be generated by the located pressure sensors are m1, Detection, and m2, No detection.Finally, the third part of the methodology is related to the formulation and solution of a single objective optimization problem, which finds the set J with the maximum VOI. Each of these parts of the methodology are detailed in the next sections and illustrated in Fig. 1, where the term Initial state of the network (S0) refers to a network without leaks (healthy system), and it is assumed that a calibrated hydraulic model is available for this condition.Part I: Leak Simulation FormulationDefinition of Leak SizesLeaks were assumed to occur in the S0 model’s nodes and were simulated as an extra demand. In the remaining of the paper, a change in demand due to the incorporation of a leak at each node will be referred to as a scenario. The demand used to simulate such leaks is called final demand (Fd), and it is a function of the base demand (Bd), the demand pattern factor (f), and the size of the leak to evaluate (Lf). Fd can be estimated using Eq. (7) (7) Lf must be large enough to cause pressure drops higher than a selected threshold (Th). In some networks, only very high leak values generate relevant pressure drops at other nodes (Pérez et al. 2009), and this in WDN-dependent. In order to generate a set of leaking values that causes observable pressure drops, the following leak value limits were defined: the lower boundary corresponds to a leak value such that a decrease in pressure bigger than Th can be detected in at least 50% of the scenarios (Lf50); upper boundary is a leak value that is the equivalent to the total WDN demand (D), representing the most critical situation. Between these two boundaries a vector V=(l1,l2,…,lk,lL) of L leaks is generated, being k the index of a particular leak in V. The increment between two consecutive leaks is obtained as (D−Lf50)/(L−1), where l1=(Lf50) and lL=D. In this paper L=10 was used.Generation of Pressure Deviation MatricesThe modeled pressure values of S0 at each node are used as reference to compare leak scenarios S. The node-to-node pressure differences between S and S0 define the pressure deviation matrices that are used to calculate two parameters, detection score (ds) and minimum leak required (MLR). The matrices consist of Pr columns, indicating the number of nodes where pressures are read, and T rows indicating the time steps simulated. The number of matrices is equal to the number of nodes where leaks are placed (Lp) multiplied by L.Definition of Detection Score and Minimum Leak RequiredThe detection score is defined as the number of times a particular node suffers a change in pressure bigger than Th along all the leak scenarios. The detection score of a particular set of sensors is estimated with Eq. (8) (8) dsJ=∑i=1Lp∑k=1LbJi,kLp×Lwhere b is defined using the following rationale: if a leak in the vector V placed at node i generates a drop of pressure with a magnitude higher than Th, in one or more nodes of the set J, then b=1; otherwise b=0.The procedure is repeated for each of the leaks in V, and at each of the Lp. From Eq. (8), it follows that a no-detection score is equal to (1−dsJ), where ds is a value ranging from zero (when the set J is not able to detect any of the lk leaks) to one (when the set J is able to detect all L leaks at all Lp nodes).Because each node requires a specific leak size to cause detectable pressure drops, it is convenient to define the MLR parameter. This is the minimum leak that needs to occur at node i such that it can be detected by a pressure sensor located at node j; this is, for each pair of nodes (i,j) there exists a minimum leak value among the L leaks evaluated that is detectable. For a set of sensors J, it corresponds to the lowest individual MLR of the sensors j that compose the set.Part II: Estimation of the Value of Information for Sensor LocationThe objective to install pressure sensors in the context of this paper is to facilitate an informed decision when there are pressure anomalies, indicating possible leaks, occur. The decision maker receives this information in the form of messages and takes an action based on the knowledge about the state of the network (prior belief), the new information (sensor messages), the available actions (decision maker’s choices), and the consequences of taking a particular decision. For the case of pressure sensor location in water distribution systems, these terms are defined as follows:Prior belief (πs) can be associated with the probability that a node experiences a leak; this probability can be estimated using field data like the node location, road conditions, surrounding trees, pipe diameters, node pressures, and pipe age, among other factors. Also, it can be associated with the decision maker’s knowledge about the system. As explained previously, the possible states s for each node are defined as Leak and No leak. Initially, the prior belief is assumed as 50% (Table 1), indicating that the decision maker does not have a preference for one of both states and is uncertain if the node is more or less prone to leaks.Table 1. Definition of the vector (πs) for two possible states of a water systemTable 1. Definition of the vector (πs) for two possible states of a water systemState, sπs (%)s1: Leak50s2: No leak50Consequences (Cas) associated with taking an action (a) given a state (s) are defined for two available actions that correspond to the possible states, a1 (Intervention) and a2 (No intervention). The costs are approximations of the expected expenditures of implementing an action, which vary according to the WDN and the characteristics of the water utility. In this methodology, a monetary unit related to the analyzed system is used. Although these are not presented for confidentiality reasons, the cost associated with the intervention action is related to the fixed costs for reparations (supplies needed and the estimated effort of the personnel required to repair leaks) and the volume of water lost. The criteria used to estimate these costs is given in Table 2 (negative values represent losses).Table 2. Definition of the Cas matrix for WDNTable 2. Definition of the Cas matrix for WDNState, sa1: interventiona2: no interventionLeak{−5−kJiif leak is detected−106if leak is not detected{p1×(−5−kJi)if leak is detected−106if leak is not detectedNo leakp2=−150For the Leak state and the action Intervention, a cost of −5−5 monetary units was assumed for fixed costs. In order to generalize the methodology, for each pair of nodes (i,j), the index k of the MLR in the vector V was selected as an indicator of the cost of water lost. When any of the leaks in V placed at node i was detected by any of the sensors in the set J, a value of 106 was used, and, as a result, the VOI of the set J against i was equal to zero (see further discussions in the “Results” section). If the decision maker decides to perform the action No intervention and the state of the system is Leak, a factor of penalization (p1) was used to increase the costs (water losses could be significant if a leak exists and it is not repaired). For demonstration purposes, p1=3 was used.For the No leak state, costs were generated only if the decision maker decided upon the action Intervention. In this case, a factor of penalization (p2) for sending a crew that could be attending other tasks was used. According to the information provided by the water utility of the region under study, it was estimated that the value of a day’s work of a crew can cost between 10 and 20 monetary units. To demonstrate the method, an intermediate value was used; therefore, p2=−15. Other values of p1 and p2 can be investigated depending on each case.Conditional probability (qms) is the probability of receiving a message m given a state s, and it is associated with the sensor’s ability to send accurate information about the state of the system. If a Detection message is always produced by a sensor when the state of the system is Leak, and if a No detection message is always produced when the state is No leak, then the information is perfect, and the matrix in Table 3 yields [1, 0; 0, 1]. Otherwise, types I and II errors are produced when No detection messages are produced in a Leak state and when Detection messages are produced in a No leak state. If this situation always occurs, the matrix in Table 3 yields [0, 1; 1, 0], and the information is totally wrong.Table 3. Definition of conditional probabilities qmsTable 3. Definition of conditional probabilities qmsState, sMessage m1: detectionMessage m2: no detectionLeakdsJ1−dsJNo leak0.150.85In the case of a Leak state, the probability of receiving a Detection message is associated with the detection score (dsJ) obtained with Eq. (8). Subsequently, the No detection message probability is associated with (1−dsJ) and corresponds to a type I error.In the case of a No leak state, a message of Detection is linked with a type II error (false alarm), where pressure variations are not associated with leaks. To facilitate the interpretation of results, the conditional probability of getting the message Detection given that the state is No leak was assumed to be 0.15, and the probability of getting the message No detection given that the state is No leak was set as 0.85. These values can be obtained using models (e.g., Alfonso and Price 2012) or can be obtained based on the accuracy stated by the sensors’ manufacturers.Once πs, qm,s, and Cas are defined, the estimation of VOI can be realized following the flowchart described in Fig. 2. The expression VOIi,J refers to the value of information that the set of sensors J provides about a leak occurring at node i.Simplified Numerical ExampleIn order to help the reader to interpret VOI for the given problem, a simplified numerical example with relevant discussions is provided. The example shows the calculation of VOI for a set of sensors J with respect to a leaking node i.If the state at node i is completely uncertain, the decision maker’s prior belief about the state of the system is Based on Table 2, and assuming that the k of the MLR to detect i was six (leak number 6 out of 10 leaks evaluated is the minimum leak placed at i that can be detected by J), the matrix of consequences is Ca,si,J=[−5−63×(−5−6)−150]=[−11−33−150]According to the prior belief and the cost matrix, the utility of each action is calculated using Eq. (1) as follows: u(a,πsi)=∑sCa,si,J×πsi=[−11−33−150]T•[0.50.5]=[−13.0−16.5]The action chosen without extra information will be the one with the maximum utility uu(a0,πsi)=max{u(a,πsi)}=max{[−13.0−16.5]}=[−13]According to this result, the action taken by the decision maker will be Intervention. The next stage is to update the decision maker’s belief with the new information generated by the set of sensors J. In this simplified example, it is assumed that the set J can generate a Detection message in 35% of cases when leaks occur; therefore, dsJ=35% and 1−dsJ=65%. According to Table 3, the values of qm,sJ are To update the prior beliefs (πs,mi,J), Eq. (3) was used, yielding πs,mi,J=qm,sJ×πsi∑sqm,sJ×πsi=[0.35×0.50.35×0.5+0.15×0.50.65×0.50.65×0.5+0.85×0.50.15×0.50.35×0.5+0.15×0.50.85×0.50.65×0.5+0.85×0.5]=[0.70.430.30.57]Results can be interpreted as follows. With πsi=[0.5,0.5], if new sensors are installed and manage to produce the message Detection, then the prior belief of state Leak is updated, increasing up to 0.7 (and in consequence, the prior belief for No leak is reduced to 0.3). If, on the contrary, the message is No detection, the prior belief of Leak is updated down to 0.43, whereas No leak is increased to 0.57. Using Eq. (4), the expected utility of each action is u(a,πs,mi)=∑sCa,si,J×πs,mi,J=[−11−33−150]T•[0.70.430.30.57]=[−12.20−13.27−23.10−14.30]Because each available action (Intervention and No intervention) has utility values now associated, the decision maker then chooses the action that gives the maximum utility for each of the possible messages using Eq. (2) u(am,πs,mi,J)=max{u(a,πs,mi,J)}=max{[−12.20−13.27−23.10−14.30]}=[−12.20−13.27]If the sensors in the set J generate a Detection message, the best decision would be Intervention (Table 4). This is because the information coming from the new sensors is able to update the prior belief of the Leak state, from 0.5% to 0.7%, and Intervention generates lower losses compared with the No intervention decision. Interestingly, when the set J generates a No detection message, the best decision is still Intervention. This is because of two reasons; firstly, the leak required at node i to be detected by J is very high (6 out 10); therefore, water loss is very high if No detection is performed. Secondly, the message comes from a sensor that 65% of the time rejects the true state (i.e., there is a leak but J does not detect it). Then, from the decision maker’s perspective, the new information is not reliable and executing the intervention is preferable, even if the sensors indicate the contrary.Table 4. Utilities of new decisions u(am,πs,mi,J)Table 4. Utilities of new decisions u(am,πs,mi,J)Actionm1: detection (by J)m2: no detection (by J)Intervention−12.20−13.27No intervention−23.10−14.30Max−12.20−13.27After the new utilities are defined, the value of each message can be obtained by comparing it with the utility of the action that would have been chosen before additional information (a0) using Eq. (5) as follows: Δmi,J=u(am,πs,mi,J)−u(a0,πsi)=[−12.20−13.27]−[−13]=[0.80−0.27]Finally, to evaluate if placing the set of sensors J in the system brings valuable information, VOI has to be calculated using the values of each individual message. Using Eq. (6), VOIi,J yields VOIi,J=∑m(∑sqm,sJ×πsi)×Δmi,JVOIi,J=[[0.350.650.150.85]T•[0.50.5]]T•[0.80−0.27]T=(0.35×0.5+0.15×0.5)×0.80+(0.65×0.5+0.85×0.5)×0.27≈0Discussion of Numerical ExampleIn the example, the set J did not generate valuable information for the decision maker about which action to take in node i once messages were generated. This is because the high leak required in node i to be detected by J and the low detection score of the set J. The initial decision before new information was Intervention, and it remained unchanged once sensors generated Detection and No detection messages. From the decision maker’s viewpoint, the extra information has no value. If the same exercise was performed with a detection score of 45%, a VOI of 0.325 would be obtained. In this case, when sensors generate a No detection message, the initial decision of Intervention changed to No intervention. This change of opinion is an indication that the new information was valuable for the decision maker. It can be concluded that a set of sensors J provides valuable information to a decision maker only if the initial decision selected without extra information changes after receiving new information.To make a general analysis of how VOI varies with different values of prior beliefs and conditional probabilities, the same exercise has been repeated, now placing two different leaks at node i: (1) a case where the lower boundary (k=1) corresponds to Lf50, and (2) the extreme case for which the node suffers the upper boundary leak D (k=10). To facilitate the interpretation, in both cases, the type II error kept at 15% (and therefore qm,s2=[0.150.85]).Fig. 3 shows the resulting variation of VOI related to changes in prior belief and conditional probabilities, Fig. 3(a) shows the VOI variation for the first case (lower-boundary leak, k=1), and Fig. 3(b) shows it for the second case (upper-boundary leak, k=10). In both cases, two characteristic lines can be recognized.The (x) line indicates a prior belief limit. Values over this limit trigger Intervention as the initial action, or the No intervention action otherwise [0.52 in Fig. 3(a) and 0.32 in Fig. 3(b)]. The value of x depends on the leak analyzed. If the leak size required in i to be identified by J is higher, there is more utility in selecting Intervention as initial decision even if the decision maker is more confident about the nonexistence of leaks at node i; this is because the higher the leaks, the greater the losses (lower utility) if No intervention is performed.The (y) line identifies the area of a perfect liar and corresponds precisely to the value of 15% assumed as false alarm or type II error. In the left side of the line (Quadrants II and III), VOIs are generated in two cases. First, the initial action changes from Intervention to No intervention when the Detection message is generated, and second, the initial decision changes from No intervention to Intervention when a No detection message is generated. In this case, the actions are opposite to the received message and occurs because the probability of generating a false alarm is higher than the probability of getting a true positive. In numerical terms, these sensors provide information, but in practice, the information emitted is completely inaccurate and they should not be installed in the network.Formulation of the Optimization ProblemIn this part, the set of sensors that provides the best quality information is selected, classifying it according to the potential for leak detection. This is the set for which VOI is maximized, and the optimization problem can be formulated as follows: Maximize:Z1=∑i=1LpVOIi,j,…,(Pdj,MLR,Ca,si,j,πsi,qm,si,j)subject to j∈PrandJ={j1,j2,,…,jn−1,jn}where  j1≠j2≠jn−1≠jnThe decision variables are the individual sensor locations j that form the set J. Because the objective function cannot be represented analytically, the use of optimization methodologies based on direct search is convenient (Marquez-Calvo 2020). In this research, a genetic algorithm (GA) was used to solve the optimization problem. The process to assign artificial leaks, estimate probabilities, obtain objective function values, and perform the optimization has been automatized by means of an algorithm framework in Python. It uses the libraries’ Water Network Tool for Resilience (WNTR) (Klise et al. 2020), which solves pressure networks using a pressure-driven approach with the solver designed in EPANET software (Rossman 2000). In addition, a Python genetic algorithm library (geneticalgorithm) (Solgi 2020) has been used to solve the optimization problem.Case StudyThe proposed methodology was applied to one WDN corresponding to a district metered area of a city in Western Europe. It was composed by 306 junctions and 254 pipes. The network was supplied through four reservoir and four pumping stations. The mean daily consumption was 20  L/s, with a minimum flow at 5:00 a.m. (6.82  L/s) and a peak flow at 10:00 a.m. (42.43  L/s). The system was composed by pipes with sizes between 40 and 1,000 mm, and the pressure without modifying the system varied from 10.86 and 21.96  mH2O. The system had four pressure sensors already installed, as shown in Fig. 4.Results and DiscussionThe methodology was applied to find the most suitable set of four sensors for the study case. To this end, the following parameters were considered: Lp=306, Pr=306, πso=[0.50.5], qm,s2=[0.150.85], Th={0.5,0.75,1.0,1.5}mH2O, and Cas as described in Table 2. For each threshold, the simulation time was set to 1 day. In addition, a time step of 1 h was set to explore all daily demand pattern values.Hydraulic ResultsDefinition of Leak Flow and Suitable NodesTo define Lf50, a range of multiple leaks were evaluated, ranging between 1.0 and 20  L/s. The objective of this step was to find the leak value such that, once added as an extra demand at each point, it generates a pressure drop greater than the Th selected at some other node in at least 50% of the scenarios.According to Fig. 5, for Th=0.5mH2O, a leak of 8.5  L/s can be identified in 50% of the scenarios, which means that 153 out 306 nodes can be detected when such a leak is added to their demand. The maximum leak evaluated (20  L/s) was detected in 75% of the cases, which means that in 25% of the nodes (76 out 306), pressure drops were lower than the evaluated Th and, therefore, were not identified by means of pressure sensors. For higher Th values, the quantity of nodes detected was reduced and the Lf50 increased. Table 5 summarizes the Lf50 values and the number of nodes that cannot be detected in each case.Table 5. Lf50 for different ThTable 5. Lf50 for different ThThreshold (mH2O)Lf50 (L/s)Nodes with drop of pressure ≥ThNodes with drop of pressure≤Th0.58.5230760.7511223831.00131931131.5017168138Fig. 6 shows the nodes that could not be detected in any of the four threshold cases. The higher the threshold value, the higher the number of nondetected nodes. This is because these nodes are close to the pumping stations, so generating pressure drops in the surrounding points requires higher leaks than the maximum leak evaluated. Additionally, each of the pumps has sufficient capacity to supply the extra flow required by the network due to leaks. For this reason, and to save computational time in the optimization procedure, the nodes that did not exhibit pressure drops were removed from Lp. This implies that this methodology is not suitable for such cases, and other methods should be used.Pressure Deviation MatrixOnce Lf50 was identified and the nondetectable nodes were removed, a list of 10 (L=10) leaks was selected (from Lf50 until 20  L/s) to run the different cases. For each case, pressure deviation matrices of size PrxT were obtained and values of MLR and k index for each pair of nodes i, j were estimated. The number of scenarios varied according to the Th selected, and ds was a function of the set of nodes J selected in the optimization process.Optimization ResultsOnce the simulations have been performed and deviation matrices obtained, the process explained in Fig. 2 was followed. As mentioned previously, the decision variables were the locations of sensors in the set of four, where each set has a respective VOI. The optimization algorithm changed the sensors within the set, and a new VOI was recalculated until a solution with the maximum VOI value was found. Because there are 306 candidate locations for sensors, and the number of proposed sensors in the set is four, there was a total of 3.58×108 possible combinations for sensor deployment. Due to this large number of possibilities, it was proposed in the optimization algorithm to explore a mutation rate of 0.1 and a crossover rate of 0.5 with a population size of 102 and a number of generations equal to 103. Thus, the search space was reduced approximately to 105. Running more generations could be explored in future research.Fig. 7 shows the optimal solution selected for Th=0.5  mH2O. In this solution, 65% of the scenarios evaluated caused a drop in pressure bigger than Th. Additionally, it is shown how each pressure sensor performed individually in terms of coverage. It shows a conveniently distributed set of sensors, such that they cover most of the network. Sensors S1 and S3 can detect most of the nodes in the network and they are complemented by S2 and S4, detecting leaks at points where the other two sensors were unable to do so.In Fig. 8, results of optimal deployment of sensors for different Th values are shown. As shown in Fig. 6, when Th varied, the amount of nodes that can be detected also changed; thus, the sensor location will be optimized to detect as many leaks as possible according to the amount of nodes that compose the search space for each case. In each of the cases, the VOI generated by the four sensors already placed (VOIsap) and the VOI for the proposed sensors (VOIsp) were obtained. The results indicated that in all cases and in VOI terms, the sensor distribution generated by the proposed methodology had a much better performance than the distribution generated by empirical practices.ConclusionsIn this paper, a methodology to optimally place sets of sensors for leak detection in WDN has been developed. It combined a pressure-driven analysis for different hydraulic simulations and the decision-making perspective about the state of the system using the concept of value of information. The method was applied in a real network with various pressure thresholds and evaluating different leak sizes. The methodology was based on the comparison of the pressures of a healthy state with the pressure deviations generated at each node in response to the incorporation of new demands (leaks), associating the occurrence of leaks to scenarios in which the pressure drops were greater than a selected pressure threshold Th. The results of the hydraulic simulation were used for the formulation of the objective functions employed in the single-objective optimization problem, in which it was proposed to maximize VOI as a measure of the quality of the information generated by the sensors. This optimization was performed by using a genetic algorithm.The use of the value of information as a parameter that includes the perception of the decision maker is very useful and appropriate because it permits to relate multiple variables to evaluate the quality of the information coming from external sources, evaluating the actions that generate the highest benefit or the lowest loss. In this way, optimal solutions can be found, adjusted to the decision makers’ requirements.In addition, it was shown that the use of multiple pressure thresholds altered the magnitude of the coverage of the network and the performance of each node in the system in leak detection. For methodologies based on the use of pressure thresholds, therefore, it is highly relevant to use multiple values of leaks and to find the appropriate size that generates the required pressure drops. The value of the pressure depends on the characteristics on the network, and for water loss scenarios, it is important that the supply sources of water such as reservoirs, tanks, or pumps are configured to represent reality in the most accurate way.The proposed methodology can be implemented for any distribution network and some suggestions for future studies are as follows: •The variation of demands at each of the nodes can be used as a parameter for comparison and selection of potential nodes for sensor location by incorporating the demand variation as a function of system pressure (PDA analysis).•Instead of evaluating the pressure change at each time step (analyzing deltas greater than a given Th), a cumulative pressure delta scheme can be considered, summing the pressure difference at each time step and evaluating if the disturbance in the system (leaks) is persistent throughout the day.•Uncertainty-aware analysis can be included to account for pressure variations due to unknown factors.•The combinatorial problem of evaluating the effect of having more than one simultaneous leak in the network can be addressed, a situation that is common in practice.Data Availability StatementAll data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request.AcknowledgmentsThis research has been supported by the European Union’s Horizon H2020 research and innovation program under Grant Agreement No. 820985 NAIADES Project.References Alfonso, L. 2010. 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