Leakage detection in water pipe networks using a Bayesian probabilistic framework

Transcription

1 Probabiistic Engineering Mechanics 18 (2003) Leakage detection in water pipe networks using a Bayesian probabiistic framework Z. Pouakis, D. Vaougeorgis, C. Papadimitriou* Department of Mechanica and Industria Engineering, University of Thessay, Pedion Areos, Voos 38334, Greece Received 17 November 2002; revised 1 Juy 2003; accepted 1 Juy 2003 Abstract A Bayesian system identification methodoogy is proposed for eakage detection in water pipe networks. The methodoogy propery handes the unavoidabe uncertainties in measurement and modeing errors. Based on information from fow test data, it provides estimates of the most probabe eakage events (magnitude and ocation of eakage) and the uncertainties in such estimates. The effectiveness of the proposed framework is iustrated by appying the eakage detection approach to a specific water pipe network. Severa important issues are addressed, incuding the roe of modeing error, measurement noise, eakage severity and sensor configuration (ocation and type of sensors) on the reiabiity of the eakage detection methodoogy. The present agorithm may be incorporated into an integrated maintenance network strategy pan based on computer-aided decision-making toos. q 2003 Esevier Ltd. A rights reserved. Keywords: System identification; Bayesian method; Leakage detection; Water pipe networks 1. Introduction Pipe networks represent one of the argest infrastructure assets of industria society. In many cases these networks suffer from aging and deterioration and fai to fufi the specified carrying capacities and required pressure heads. The high maintenance and revamping costs, incuding rehabiitation, repacement and/or expansion of existing systems to meet current and future demands give rise to difficut decision-making. A these activities, due to the arge amount of money to be invested, usuay become of primary pubic interest. One of the major probems to be faced is the frequent pipe-breaks with unaccounted water eakages resuting in service disruption. Water service companies have begun to deveop new eakage detection strategies in order to reduce eakages to an economica optimum eve [1]. The main objective is to propose reiabe computationa modes to faciitate pipe repacement decisions in an effort to increase the overa reiabiity expected from the pipe network. An extensive amount of work on pipe rehabiitation and repacement has been pubished. The various agorithms * Corresponding author. Te.: þ ; fax: þ E-mai address: costasp@mie.uth.gr (C. Papadimitriou). deveoped have taken the form of non-inear, dynamic, heuristic and successive inear programming economic modes, which assist decision-making based usuay on historica statistics and cost information. In an eary work Shamir and Howard [2] proposed a mode, which estimates the optima time for pipe repacement based on pipe breakage history and the cost for repairing and repacing pipes. Ketter and Gouter [3], identified a reationship between breakage rate and pipe diameter as we as a correation between the number of pipe faiures and pipe age. They proposed that improvements to pipe breakage or mechanica reiabiity may be achieved by seecting arger pipe diameters. Woodburn et a. [4] presented a mode for determining the minimum cost for rehabiitation, repacement or expansion of an existing network based on a combination of non-inear optimization and hydrauic simuation procedures. An expicit agorithm, impementing a graph theory approach, has been deveoped by Bouos and Atman [5]. The agorithm is capabe of handing widespread appications, associated with future panning, expansion and improvement of fuid distribution networks. Aruraj and Rao [6] proposed an optimaity criterion caed the significance index to rehabiitate existing networks. On many occasions when continuous quantities are seected as decision variabes the resuts may be miseading /$ - see front matter q 2003 Esevier Ltd. A rights reserved. doi: /s (03)

2 316 Z. Pouakis et a. / Probabiistic Engineering Mechanics 18 (2003) since pipes are coming in discrete engths and diameters. Kim and Mays [7] resoved the probem, to some extent, using integer pipe engths as decision variabes. More recenty an increasing number of researchers are impementing genetic agorithm techniques in certain aspects of the design and rehabiitation of pipe networks, e.g. Murphy et a. [8] and Simpson et a. [9]. Over the years genetic agorithms have proven to be a reiabe technique for handing water distribution network probems. In addition to their abiity to hande discrete pipe diameters they have been shown to be quite robust and efficient in searching for optima rehabiitation poicies [10,11]. Foowing a these efforts water distribution companies have started atey, very reuctanty however, to impement these computationa approaches and corresponding software as decision-making toos in the management of their water networks [1]. An issue, which is ceary reated to an efficient eakage reduction poicy but which has received much ess attention, is the on-ine eakage identification. Most of the research works performed and discussed above are not focused on eakage detection. It is mosty reated to the genera issue of deveoping efficient agorithms eading to optima repacement, rehabiitation or expansion soutions for pipe networks. Rea-time damage estimation and diagnosis of buried pipeines, however, pays an important roe in an integrated maintenance network strategy. Leakage detection may be considered, of course, as part of a typica caibration procedure and can be handed via optimization agorithms using conventiona and evoutionary approaches. It is important, however, that the proposed agorithm is capabe of incorporating in a quantitative manner a the errors in the mode compared to the rea probem. As reported by Gouter and Bouchart [12], very itte research work has been reported on the incusion of these probabiistic issues in optimization design modes for water distribution networks. It is hepfu to have a methodoogy to convert a these uncertainties and errors introduced in water pipe network optimization modeing into a measurement of the reiabiity of the resuts obtained by the modeing procedure. Here we address this issue focusing on eakage identification agorithms. Leakage can be detected by correating changes in fow characteristics to changes in a hydrauic mode for the network. Significant changes in the hydrauic mode are indicative of the ocation and the severity of the damage. This correation is achieved by updating the hydrauic mode so that its predictions match the measured data obtained from the sensors. This mode updating procedure is an inverse probem that is usuay i-conditioned due to ack of sensitivity of the fow characteristics to modest amounts of eakages, and often non-unique due to insufficient avaiabe data reative to the mode (network) compexity. Difficuties associated with the deveopment of effective mode updating and eakage detection agorithms are: (1) modeing errors (difference between theoretica mode and actua system behavior), (2) measurement errors, and (3) incompete set of observed data due to imited number of sensors avaiabe or due to imited accessibiity within the network. Very recenty Shinozuka and Liang [13] deveoped an approach to identify the ocation and the severity of damage in a water deivery system by monitoring on-ine water pressures at some seected positions of the network. Their damage detection approach is based on a neura network inverse anaysis method. Aso Andersen and Powe [14] proposed a eak detection scheme based on an impicit formuation of the standard weighted-east-squares stateestimation probem. In both cases however, the schemes are appied to ideaized noise-free conditions. In the present work a Bayesian system identification methodoogy is proposed for mode updating which aows for the expicit treatment of the uncertainties arising from modeing errors and measurement noise. The methodoogy has been we deveoped and successfuy appied in structura mode updating appications [15 18]. Here, the Bayesian methodoogy is modified accordingy and it is couped with hydrauic simuations for updating a parameterized cass of hydrauic modes with the parameters chosen to simuate a set of possibe eakage events (ocation and severity of eakage) in the pipe network. The methodoogy provides estimates of the probabiity of each eakage event (eakage ocation and severity) given the fow and/or pressure head measurements obtained from an integrated monitoring and data management system set up for the network. The most probabe eakage event is identified as the one with the highest probabiity, whie the other eakage events are ordered according to their reative probabiities. The effectiveness of the proposed framework is iustrated by appying the eakage detection approach to a specific water pipe network. Severa important issues are addressed, incuding the roe of modeing error, measurement noise and sensor configuration (ocation, number and type of sensors) on the reiabiity of the eakage detection methodoogy. 2. Formuation A typica hydrauic formuation is used for the soution of the water pipe network. The fow equations to be soved consist of the mass conservation equations at the junction nodes and the energy conservation equations around the oops and the pseudo-oops of the network [19]. The system is soved using a Newton iteration scheme. Once the pipe fow rates are estimated the energy grade at the nodes is expicity estimated through a marching procedure. Leakage detection is based on the premise that damage (eakage) in one or more ocations of the piping network invoves oca iquid outfow at the eakage ocation, which wi change the fow characteristics (pressure heads, fow rates, acoustics signas, etc.) at the monitoring ocations of the piping network. The magnitude of

3 Z. Pouakis et a. / Probabiistic Engineering Mechanics 18 (2003) changes in the fow characteristics depends on the position and the severity of the damage (amount of outfow). The existence of eakage in the pipe network is diagnosed by monitoring the permanent changes in the fow characteristics of the system. Once the existence of the damage has been diagnosed, it is possibe by updating the hydrauic mode for a compete set of parameters to identify the ocation and the severity of the damage. For this, a statistica system identification methodoogy is appied that effectivey tackes the uncertainties due to modeing error and measurement noise Statistica system identification The hydrauic pipe network formuation is impemented to generate a cass of soutions describing the fow behavior of the piping distribution system in an undamaged state or in a damaged (deteriorated) state due to eakage events. This cass of soutions is generated from a parameterized cass of modes denoted by M: Let u be the parameters introduced in the parameterized cass of hydrauic modes M: In the case of eakage identification these parameters are associated with the ocation and extend of the damage in the piping distribution system. A particuar mode MðuÞ from the cass M is seected by specifying the vaues of the parameter set u: Next consider a monitoring system that has been instaed in the network in order to coect and anayze data obtained from N fow tests performed periodicay at L monitoring ocations. Let ~x ij be the avaiabe fow data from the jth fow test ð1 # j # NÞ obtained at the ith monitoring ocation ð1 # i # LÞ: Without oss of generaity we may assume that the fow data set may consist of pressure heads ~P and fow rates ~Q estimates obtained at the L monitoring ocations. Let ~x ¼½~x ij Š denote the L N matrix of data at L ocations from N fow tests. Let aso xðuþ ¼½x ij ðuþš be the fow quantities (pressure heads, fow rates, etc.) at the monitoring ocations computed from the mode MðuÞ corresponding to a particuar vaue assigned in the parameter set u: The departure between the mode resuts and the corresponding measured fow quantities, defined by e ij ðuþ ¼~x ij 2 x ij ðuþ ð1þ measures the prediction error from the mode MðuÞ: This departure is due to fow network modeing error and device measurement accuracy that are unavoidabe in the modeing process of rea water distribution systems. System identification is handed by empoying a statistica approach [15,20] in which the mode prediction error e ij ðuþ is considered to be a specific reaization of a random variabe taken from a cass of probabiistic error modes P; parameterized by the parameter set s: The cass of pipe fow modes M and the cass of prediction error modes P; which specify the modeing assumptions used in the description of the system, are parameterized by the parameter set ½u; sš: The objective of the statistica system identification methodoogy is to update the vaues of the parameter set ½u; sš and their associated uncertainties using the measured test data. Here, uncertainty in the vaues of the parameter set is quantified using probabiity density functions (PDF), which measure the reative pausibiity of each of the modes in casses M and P specified by the parameters ½u; sš: The seection of the parameter uncertainty prior to the coection of data is based on engineering experience and it is quantified by the initia PDF pðu; sþ: Using Bayes theorem, this initia PDF is converted to a posterior (updated) PDF pðu; s~xþ ¼c 1 pð~xu; sþpðu; sþ ð2þ which gives the reative pausibiity of the modes based on the incusion of the measured data ~x: The constant c 1 is a normaizing constant seected such that the posterior PDF pðu; s~xþ is integrated to one. In Eq. (2) the expression for the posterior PDF pðu; s~xþ depends on the chosen casses of pipe network modes M; the prediction error modes P and the measured data. Assuming e ij ðuþ; i ¼ 1; ; L; j ¼ 1; ; N to be independent and normay distributed with zero mean and standard deviation s; the ikeihood pð~xu; sþ may be written in the form " # pð~xu; sþ ¼ YL Y N i¼1 j¼1 1 pffiffiffiffi exp 2 ðx ijðuþ 2 ~x ij Þ 2 2p s 2s 2 : ð3þ Assuming aso a non-informative prior distribution for the mode parameters over the range of acceptabe vaues of ½u; sš; i.e. assuming that the initia PDF pðu; sþ is constant, and substituting Eq. (3) into Eq. (2) yieds 2 3 X N kx j ðuþ 2 ~x j k 2 1 pðu; s~xþ ¼c 2 pffiffiffiffi ð 2p exp 2 j¼1 sþ LN 6 4 2s 2 ð4þ 7 5 where k k is the usua Eucidean norm. The vector x j ðuþ denotes the mode resuts, whie the vector ~x j denotes the corresponding measured fow quantities at the measured ocations from the jth fow test. The constant c 2 in Eq. (4) is seected such that the posterior PDF pðu; s~xþ is integrated to one Optima mode and mode uncertainty The optima vaue of the mode parameters denoted by ½ ^u; ^sš; is simpy the most probabe vaue of ½u; sš obtained by maximizing the updated PDF pðu; s~xþ or equivaenty by minimizing the function gðu; sþ ¼2n½pðu; s~xþš ¼ 1 2s 2 X N i¼1 k~x i 2 x i ðuþk 2 þ LN 2 n s2 þ c: ð5þ

4 318 Z. Pouakis et a. / Probabiistic Engineering Mechanics 18 (2003) The optima ½ ^u; ^sš for a given sensor configuration depends ony on the data ~x: Performing the optimization for a set of continuous parameters u; it can be readiy shown that the optima vaues ^u of the network mode minimize the function JðuÞ ¼ 1 X N kx LN j ðuþ 2 ~x j k 2 ð6þ j¼1 where JðuÞ represents a norm of the difference between the mode and the measured output. Through the optimization procedure it can aso be shown that the optima vaue ^s 2 of the prediction error mode is ^s 2 ¼ Jð ^uþ: In particuar, using the tota probabiity theorem, the margina probabiity distribution for u is obtained as ð pðu~xþ ¼ pðu; s~xþds: ð8þ Substituting Eq. (4) into Eq. (8) and after some mathematica manipuation, using Eq. (6) and assuming a uniform initia distribution pðu; sþ the integration in Eq. (8) is carried out anayticay to yied pðu~xþ ¼c½JðuÞŠ 2ðLN21Þ=2 : For a genera initia distribution pðu; sþ an asymptotic approximation is avaiabe in the form [21] pðu~xþ ¼c½JðuÞŠ 2ðLN21Þ=2 gðuþ ð10þ pffiffiffiffiffi where gðuþ ¼pðu; JðuÞÞ: Expression (9) or (10) yieds the uncertainty in the optima estimate of the mode parameters given the measured data Appication to eakage detection Consider the case of network deterioration due to the presence of fractures in one or more ocations of a water pipe network consisting of p pipes. Such events wi invove oca water outfows that can be modeed by adding a fow demand at each fractured pipe indicating the ocation of each eakage. In addition the amount of each fow demand wi correspond to the severity of the eakage in that ocation. Let K be the number of eakage ocations. The amount of fow demands and the eakage ocations (pipe ocations where these demands are added) constitute the set of unknown parameters u of the integrated cass of modes M describing the behaviour of the system with eakages. The foowing mode parameterisation can be adopted to efficienty identify the eakage in the network assuming that eakage occurs at K ocations. The mode parameter set u is partitioned into two subsets written as u ¼ðu ; u s Þ: The subset u is a K-dimensiona vector of integers denoting the pipe sections that have eakage, i.e. denoting the ocations of eakage in the network. The tota number of distinct eakage ð7þ ð9þ events in a water distribution system with p pipe sections and K eakage ocations is p! N K ¼ K!ðp 2 KÞ! : ð11þ The subset u s is aso a K-dimensiona vector giving the amount of iquid outfow, quantifying the eak severity (amount of eakage) at the corresponding K eakage ocations identified in the set u : The optima vaues ^u ¼ð^u ; ^u s Þ of the mode parameters u ¼ðu ; u s Þ are computed by maximizing expression (9). This optimization probem invoves a mixed set of discrete and continuous variabes. The discrete variabes, incuded in the parameter subset u ; take integer vaues from 1 to p; indicating the number of the pipe that has eaked, whie the continuous variabes, incuded in the parameter subset u s ; can ony take positive vaues since eakage invoves iquid outfow. The soution scheme that is adopted to sove the optimisation probem with K eakages invoves an exhaustive search over the discrete parameter subspace. Specificay, et u ðiþ denote the eakage ocations corresponding to the ith eakage event taken from the tota of N K distinct ðiþ eakage events. The most probabe vaue ^u s of the parameter set u s ; given that eakage occurs at ocations u ðiþ ; are obtained by maximizing the PDF, given by expression (9), with respect to the parameters in the set u s : This is a continuous optimisation probem invoving the K parameters in the set u s : Then the most probabe eakage event ^u is the one of u ð1þ ; u ð2þ ; ; u ðkþ that maximizes the updated PDF function pðu ðiþ ðiþ ; ^u s ~xþ ¼c½Jðu ðiþ ðiþ ; ^u s ÞŠ 2ðLN21Þ=2 : ð12þ Thus the most probabe eakage event ^u ¼ð^u ; ^u s Þ can be obtained by soving a series of N K optimization probems. An exhaustive search of the most probabe eakage event requires the soution of N K optimization probems which may be computationay expensive or even prohibitive, considering that each function evauation of xðuþ invoved in Eq. (9) requires the soution of a non-inear agebraic system of equations governing the steady-state fow in pipe networks. Instead, genetic agorithms [22] can be used to efficienty sove this type of discrete optimization probem in order to provide a near optima soution for the eakage ocations. In practice, it is expected that deterioration wi proceed progressivey with eakage occurring at one ocation at a time. Using a monitoring system, the agorithm coud be used to search for singe eakages. This invoves the soution of as many as N K ¼ p optimization probems and thus the computationa effort is reativey sma. Furthermore, when the eakage ocations are expected to occur ony in a certain number of pipes forming the network the computationa compexity of the probem is significanty reduced, even for the case of searching for mutipe eakage ocations.

5 Z. Pouakis et a. / Probabiistic Engineering Mechanics 18 (2003) To concude the theoretica formuation of the methodoogy, it is stated that the eakage detection in networks with mutipe eaking events invoves a continuous optimization probem searching for the eakage severity, which is embedded in a discrete optimization probem searching for the most probabe eakage ocation. 3. Appication to networks In order to demonstrate the effectiveness and efficiency of the proposed eakage detection methodoogy the whoe approach is appied to an exampe network shown in Fig. 1. This network can be considered as a simpified, typica, municipa water distribution system or a water distribution network for an industria unit. It comprises 50 pipe sections, 31 junction nodes and 20 oops (no pseudo-oops). The water is suppied from an eevated tank by gravity. The engths of the horizonta and vertica pipe sections are 1000 and 2000 m, respectivey, and the eevation of the tank is 52 m. Pipe and junction numbering is given in Fig. 1, where the pipe diameters, varying from 300 to 600 mm, are aso indicated. Fow demands are assumed at each node of the network throughout the water deivery system. The water fow is in the turbuent fow regime with a friction factor estimated by the formua of Swamee and Jain [19]. The cass of modes used for identifying eakage in the network assumes that the piping-roughness coefficients are the same for a pipes and the fow demands are uniform throughout the water deivery system. The nomina vaues for the piping-roughness coefficients are taken to be equa to 0.26 mm for a pipes and the fow demands are assumed equa to 50 /s at each junction node. The tota voume of water suppied from the eevated tank is 1550 /s and is equa to the tota fow demands in the network Data simuation Measured data are simuated from a pipe network mode with characteristics that are different from the ones that correspond to the cass of modes used for monitoring and identification of the network condition. The measured data produced in this way aows the simuation and study of the mode error effect on the eakage detection resuts. In addition, in order to account for the measurement noise in the sensors, an error term is added to the predictions of the perturbed mode to simuate the observed discrepancy between the actua pipe network predictions and the measurements from the sensors. Specificay, the simuated measured data are generated from the foowing equation ~x ij ¼ ~x m ij þ ~x n ij; 1 # i # L; 1 # j # N: ð13þ The first term, ~x m ij ; in Eq. (13) represents the pressures and/or the fow rates that are generated from a cass of pipe network modes with characteristics that deviate from the nomina characteristics defining the cass of modes used for identification. Here, the characteristics that are perturbed from their nomina vaues incude the piping roughness coefficients in each pipe and the fow demands at the nodes. The perturbation from the nomina vaues of the pipingroughness coefficients for each pipe and of the fow demands for each node is assumed to foow a zero-mean uniform distribution with bounds ð2a; aþ and ð2b; bþ; respectivey. The size of the perturbation, that is, the vaues of a and b; represents the magnitude of the mode error expressed as a percentage of the nomina vaues of the system characteristics. The statistica generation of these perturbations refects the uncertainty in the actua vaues of these parameters. Hereafter, the aforementioned perturbed mode is assumed to be representative of the actua behaviour of the system and is referred to as the actua system. Fig. 1. Water pipe network configuration.

6 320 Z. Pouakis et a. / Probabiistic Engineering Mechanics 18 (2003) It is obvious that when the above procedure to simuate the measured data is used, the cass of modes for identification is not capabe of representing the behaviour of the actua system exacty. Foowing this approach it is possibe to simuate and study the effects of the mode error on the eakage detection resuts. The second term, ~x n ij; in Eq. (13) accounts for the measurement error that comes from the sensors. It is chosen to be a zero-mean, uniformy distributed random variabe with bounds ð2c; cþ: The magnitude of c represents the size of measurement error at a measured ocation expressed as a percentage of the actua system predictions at the measured ocation. Depending on the type of the measured data used in the identification procedure, two cases, namey Cases A and B, are considered separatey. In Case A the measured data consists of pressures at junction nodes, whie in Case B the measured data consists of fow rates in pipes. For demonstration purposes, the case of a singe eakage and the case of two eakages are examined separatey Detection of a singe eakage The case of a singe eakage ðl ¼ 1Þ ocated at pipe section 26 is considered. Seven monitoring devices are used, spread a over the network and reativey far from the vicinity of the eaked pipe 26. For Case A the manometers are ocated at the junction nodes 6, 9, 13, 18, 21, 25 and 30, whie for Case B the fow meters are ocated at the pipe sections 1, 3, 16, 19, 29, 36 and 43. The eakage ocation is simuated by adding a node at the damaged pipe whie the prescribed fow demand corresponds to the amount of eakage at the eakage ocation. Simuated measured data are generated from Eq. (13) using the actua system with the corresponding eakage ocation. The probem of identifying the eakage ocation and severity is addressed given that ony a singe eakage is expected. The number of possibe eakage scenarios, given in Eq. (11), is equa to the number of pipe sections in the water network ðn K ¼ 50Þ: The cass of modes used for eakage identification invoves two parameters ðu ; u s Þ; with u denoting the eakage ocation and u s accounting for the amount of eakage. By means of the identification procedure, first the most ^u ðiþ s probabe amounts of eakage are estimated for each eakage scenario i: In the case of the singe eakage, i indicates the pipe number that has eaked. Then the normaized probabiity p i ¼ kpðu ðiþ ðiþ ; ^u s ~xþ of the parameter set u ðiþ s ð14þ of each eakage event i is computed, for the corresponding ðiþ most probabe eakage severity vaue ^u s : The vaues of p i estabish an order for the eakage events according to their reative normaized probabiity. The most probabe eakage event is identified as the one with the highest normaized probabiity p i : The normaizing constant, k ¼ P N K i¼1 pðuðiþ ðiþ ; ^u s ~xþ; is convenienty used for potting purposes without affecting the interpretation of the resuts Ideaized scenario with no errors First, no mode or measurement uncertainties are introduced ða ¼ b ¼ c ¼ 0%Þ: The amount of eakage is taken to be equa to 22.8 /s, which corresponds to 1.5% of the tota water voume suppied in the network. The computed peak vaues of the normaized posterior PDF p i are potted in Fig. 2A and B for each pipe section of the water distribution network for Cases A and B, respectivey. It is seen that in both cases tested, athough the eakage severity is quite sma compared to the tota water voume, the proposed methodoogy identifies correcty the pipe where the eakage is assumed. When no mode or measurement uncertainties are introduced, the peak vaue of the posterior PDF of the most probabe scenario is equa to one, whie the corresponding vaues for a other Fig. 2. Peak vaues of normaized PDF at each pipe section using (A) manometers and (B) fow meters. Leakage is ocated at pipe 26 with severity equa to 22.8 /s (1.5% of the tota water voume).

7 Z. Pouakis et a. / Probabiistic Engineering Mechanics 18 (2003) Fig. 3. Peak vaues of normaized PDF at each pipe section using (A) manometers and (B) fow meters. Leakage is ocated at pipe 26 with severity equa to 22.8 /s (1.5% of the tota water voume). A perturbation a ¼ 5 and 10% is assumed in the piping roughness coefficient. undamaged pipes are identicay equa to zero (within round-off error), independenty of the eakage severity. This is a typica resut for a cases tested, and in this way the assumed ocation and severity of the eakage is easiy found Effect of mode errors The above situation is highy ideaized and amost never occurs in rea water deivery systems. More reaistic situations are simuated next, when some mode errors are introduced by imposing perturbations in the vaues of the roughness coefficient and the fow demands. The amount of eakage is taken to be equa to 22.8 /s. First, perturbations equa to 5 and 10% are introduced in the mode error magnitude a; corresponding to the piping roughness coefficient. The resuts reated to the identification of the eakage ocation are potted in Fig. 3A and B using manometers (Case A) and fow meters (Case B), respectivey. The most probabe eakage scenario predicts eakage in pipe 26, which coincides with the actua eakage ocation. Again, the peak vaues of the posterior PDF of the most probabe eakage event are severa orders of magnitudes arger than the corresponding vaues of the other eakage scenarios. This is an indication that the ocation of the damaged pipe is ceary identified even for sma amounts of eakage compared to the nomina fow rate of the particuar pipe and for reativey arge mode error introduced in the parameterized mode. The computed amounts of eakage are compared to the actua amounts in Tabe 1 for Cases A (Figs. 2A and 3A) and B (Figs. 2B and 3B). It is seen that when a ¼ 0% the actua amounts of eakage are exacty estimated. For a ¼ 5 and 10% there is some departure between the computed and the actua amounts when the experimenta data are obtained using pressure-measuring devices. The higher the mode error is, the higher is the departure between the identified and the actua amounts of eakage. The agreement is exceent when fow meters are used. We continue our study on the effects of the mode uncertainties on the eakage identification process by impementing a second mechanism to introduce mode errors. The nomina vaues of the fow demands are perturbed by an amount b equa to 2 and 5%. Fig. 4A and B shows the corresponding normaized peak vaues of the posterior PDF in each pipe for Cases A (manometers) and B (fow meters), respectivey. It is seen that the departure of the peak vaues of the posterior PDFs between the most probabe scenario and the second most probabe scenario becomes smaer compared to corresponding previous resuts (Fig. 3A and B). It is pointed out however, that the most probabe eakage scenario coincides with the actua eakage event. Even for a sma eakage equa to 1.5% of the tota water suppy and for an uncertainty equa to 5%, the departure between the most and the second most probabe events is at east one order of magnitude. It is worth noting that the second most probabe eakage scenario corresponds to eakage at pipe 21, which is adjacent to pipe 26. It shoud be noted, however, that for even more difficut situations invoving even smaer amounts of eakage and/or arger uncertainties (arger mode errors) it wi not be possibe to identify successfuy the eakage ocation. There is aways a threshod eve beyond which no reiabe resuts are obtained. The proposed methodoogy is capabe of identifying these threshod vaues. In many cases when the accurate eakage ocation is not possibe it is sti usefu to Tabe 1 Comparison between computed and actua amounts of eakage in pipe 26 for various uncertainties in the pipe roughness coefficient using (A) manometers and (B) fow meters Case Actua eakage (/s) Computed eakage (/s) a ¼ 0 a ¼ 5% a ¼ 10% A B

8 322 Z. Pouakis et a. / Probabiistic Engineering Mechanics 18 (2003) Fig. 4. Peak vaues of normaized PDF at each pipe section using (A) manometers and (B) fow meters. Leakage is ocated at pipe 26 with severity equa to 22.8 /s (1.5% of the tota water voume). A perturbation b ¼ 2 and 5% is assumed in the demands. identify the region in the pipe network which contains the damaged pipe. In Tabe 2 the computed amounts of eakage for Cases A (Figs. 2A and 4A) and B (Figs. 2B and 4B) are compared to the actua amounts. It is seen that the eakage severity for b ¼ 0 is computed exacty, whie for b 0 it is overestimated. It is evident that for the water pipe network under investigation the estimation of eakage severity is particuary sensitive to demand variations. In these cases, an improved sensor pacement configuration capabe of coecting better information from the network is essentia and wi resove to some extend the probem. It is seen that, for the pipe network under examination, the identification of the amount of eakage is ess accurate than the correct identification of the eakage ocation when uncertainties in the demands are introduced in the sense that this identification is more sensitive to mode errors. In genera, as expected, the departure between computed and actua quantities becomes arger as the eakage severity is decreased and as the mode errors are increased Effect of measurement errors The study on the effects of the errors is competed by impementing additiona uncertainties due to measurement errors. These errors are again simuated by adding a zeromean uniform noise in the data generated by the actua system, with a standard deviation c equa to 2 and 5% of the actua system predictions at the measured ocations. Two eakage severity scenarios are considered which correspond to eakage amounts equa to 57.0 and 22.8 /s, which represent 3.7 and 1.5%, respectivey, of the tota suppied water voume (1550 /s). In Fig. 5A and B the computed peak vaues of the posterior PDF are potted for each eakage scenario and for each pipe section of the water distribution network for Cases A and B, respectivey. For c ¼ 2% the eakage ocation is correcty identified for a cases. It is worth noting that the second most probabe eakage ocation predicted by the methodoogy is pipe 20 or 21, which is very cose to the correct eakage ocation. For arger measurement errors ðc ¼ 5%Þ the eakage ocation is correcty identified for the cases of 57.0 /s. However, for the smaer eakage of 22.8 /s, the correct eakage ocation is not identified in Case A. It is interesting to note that this sma amount of eakage corresponds to 1.5% of the tota water suppy. Moreover, in Case B, athough the correct eakage ocation is identified as the most probabe, other eakage ocations have been predicted with a probabiity cose to that. In addition, in Tabe 3 the computed amounts of eakage for Cases A (Figs. 2 and 5) and B (Figs. 2B and 5B) are compared to the actua amounts. For c ¼ 0; the correct amount of eakage is ceary identified and the exact amount of eakage is computed. It is aso noted that for c 0 the amounts of eakage, as is shown in Tabe 3, are computed with good accuracy for a cases tested. It may be concuded that for the water pipe network under investigation the errors in the experimenta data have a more serious effect on the sensitivity of the resuts reated to the ocation rather than the severity of the eakage Effect of sensor type and ocation We concude this section by investigating the effect of the positioning of the measuring devices on the system identification methodoogy. This is demonstrated by studying the same test case network shown in Fig. 1. The eakage Tabe 2 Comparison between computed and actua amounts of eakage in pipe 26 for various uncertainties in the demands using (A) manometers and (B) fow meters Case Actua eakage (/s) Computed eakage (/s) b ¼ 0 b ¼ 2% b ¼ 5% A B

9 Z. Pouakis et a. / Probabiistic Engineering Mechanics 18 (2003) Fig. 5. Peak vaues of normaized PDF at each pipe section using (A) manometers and (B) fow meters. Leakage is ocated at pipe 26 with severity equa to (i) 57.0 and (ii) 22.8 /s (3.7 and 1.5% of the tota water voume). A perturbation c ¼ 2 and 5% is assumed in the modeed measurements. is kept in pipe 26 and ony the case of eakage severity equa to 57.0 /s, i.e. 3.7% of the tota water suppy is studied. The seven measuring devices, however, are paced in different positions. This new positioning, which can be either ess or more informative about the system is such that the sensors are concentrated in certain sub-regions of the network. The manometers (Case A) are a ocated in the ower right region of the network, specificay at junctions 17, 18, 19, 23, 24, 25 and 31. The fow meters (Case B) are ocated in the upper eft region of the network, specificay at the pipe sections 1, 2, 3, 7, 18, 25 and 26. The fow meter configuration incudes a sensor, which is ocated at the eaking pipe 26. In Fig. 6A and B the computed normaized peak vaues of the posterior PDF are potted for each pipe section of the water distribution network using manometers (Case A) and fow meters (Case B), respectivey. A three different types of uncertainties invoving the pipe roughness coefficients a; the fow demands b and the measurement data c are examined. Consequenty, a direct comparison between the present resuts and those obtained with the previous measuring devices configuration is possibe. Comparison of the resuts in Figs. 3 and 6(i) deaing with uncertainties in pipe-roughness coefficients shows that with the new sensor configuration the identification resuts are improved for the case of fow measurements, whie they have deteriorated for the case of pressure measurements. Comparison of the resuts in Figs. 4 and 6(ii) deaing with uncertainties in fow demands shows that the identification resuts obtained from the new sensor configuration are ess informative for both the cases of fow and pressure measurements. In particuar, Tabe 3 Comparison between computed and actua amounts of eakage in pipe 26 for various uncertainties in the measurement data and two different amounts of eakage using (A) manometers and (B) fow meters Case Actua eakage (/s) Computed eakage (/s) c ¼ 0 c ¼ 2% c ¼ 5% A B

10 324 Z. Pouakis et a. / Probabiistic Engineering Mechanics 18 (2003) Fig. 6. Peak vaues of normaized PDF at each pipe section using (A) manometers in the nodes (17, 18, 19, 23, 24, 25, 31) and (B) fow meters in the pipe sections (1, 2, 3, 7, 18, 25, 26). Leakage is ocated in pipe 26 with severity equa to 57.0 /s. the methodoogy fais to predict the actua eakage ocation for the case of b ¼ 5%: The most probabe eakage ocation predicted by pressure measurements is in pipe 20 and in pipe 27 for fow rate measurements, both of which are adjacent to the damaged pipe 26. Finay comparison of the resuts in Figs. 5(i) and 6(iii) deaing with uncertainties in the measured data shows that the identification resuts obtained from the new sensor configuration are significanty improved for both cases of pressure and fow rate measurements. Specificay, in contrast to the previous

11 Z. Pouakis et a. / Probabiistic Engineering Mechanics 18 (2003) Tabe 4 The first five most probabe eakage events with a perturbation a ¼ 5% in the pipe roughness coefficient using (A) manometers and (B) fow meters. Actua eakage ocations are in pipes 26 and 42 with corresponding amounts 114 and 44.7 /s Case Leakage ocation (pipe number) Leakage severity (/s) Peak vaue of normaized PDF Location 1 Location 2 Severity 1 Severity 2 A B sensor configuration, which was not informative enough for accurate eakage predictions, the new sensor configuration provides reiabe prediction of the position of eakage. The above resuts indicate ceary that an optima sensor pacement strategy [23,24] is very important for efficient detection of damage in water pipe networks. The critica issue is to estimate the proper number and ocation of sensors in order to obtain the maximum possibe information from the network without knowing in advance the ocation and amounts of eakage in the system. This is a difficut task which wi be addressed in detai in ater work Detection of mutipe eakages It has been shown in Section 2 that the proposed methodoogy is capabe in a straightforward manner of studying simutaneous mutipe eakages in a water distribution system. An appication is presented in this section using the water pipe network of Fig. 1, with two eakages in pipe sections 26 and 42. The amounts of eakage are taken to be 114 and 44.7 /s, which represent 7.4 and 2.9%, respectivey, of the tota water suppy. It is noted that whie in the case of one eakage there are 50 possibe eakage scenarios, now, with two eakages, it is estimated using Eq. (11) with K ¼ 2 and p ¼ 50 that there are N k ¼ 1225 possibe eakage scenarios. It is obvious that the required computationa effort is significanty increased. The impementation however of the whoe approach remains the same. The resuts from the identification methodoogy are very simiar to the case of one eakage event. When no uncertainties exist in the system, the resuts indicate the two eakage ocations and the corresponding amounts very accuratey, independenty of the positions and the amounts of eakage. When uncertainties are introduced, more carefu investigation is required, since there are severa eakage scenarios with a probabiity vaue cose to the probabiity of the most probabe eakage scenario. Some typica resuts from the most probabe eakage scenarios are presented in tabuated form. In Tabe 4, the first five most probabe eakage events are given based on the use of manometers (Case A) and fow meters (Case B) with introduced uncertainty a ¼ 5% in the pipe roughness coefficient. Each eakage event is described in terms of Tabe 5 The first five most probabe eakage events with a perturbation b ¼ 2% in the demands using (A) manometers and (B) fow meters. Actua eakage ocations are in pipes 26 and 42 with corresponding amounts 114 and 44.7 /s Case Leakage ocation (pipe number) Leakage severity (/s) Peak vaue of normaized PDF Location 1 Location 2 Severity 1 Severity 2 A B

12 326 Z. Pouakis et a. / Probabiistic Engineering Mechanics 18 (2003) Tabe 6 The first five most probabe eakage events with a perturbation c ¼ 2% in the measurement data using (A) manometers and (B) fow meters. Actua eakage ocations are in pipes 26 and 4 with corresponding amounts 114 and 44.7 /s Case Leakage ocation (pipe number) Leakage severity (/s) Peak vaue of normaized PDF Location 1 Location 2 Severity 1 Severity 2 A B the two-eakage ocation and severity and its corresponding normaized probabiity. The probabiity resuts shown in the ast coumn have been normaized such that for each case the most probabe eakage event corresponds to probabiity equa to one. Simiar resuts are given in Tabes 5 and 6 for introduced uncertainties in the demands ðb ¼ 2%Þ and in the experimenta data ðc ¼ 2%Þ; respectivey. It is seen that in most cases the actua eakage event is estimated correcty. Again it has been found that as the amount of eakage is reduced and the mode and measurement errors are increased there are certain threshod vaues of mode and measurement errors above which the identification of the eakage event is not possibe. The resuts in Tabes 4 6 have been obtained using a combinatoria approach searching for a possibe eakage scenarios. For more than two eakages, genetic agorithms [22] may be impemented in order to sustain reasonabe computationa effort and time eves converging to the near goba soution. 4. Concusions A Bayesian probabiistic framework for eakage detection in water pipe networks has been deveoped and successfuy tested with simuated data. Prediction of the most probabe eakage ocations and severity invoves a series of continuous optimization probems foowed by a discrete optimization probem. For the cases considered, an exhaustive search is used to sove the discrete optimization probem. When the system is free from mode error and measurement noise, the mode is capabe of identifying the damage exacty. More reaistic circumstances are aso examined in detai by introducing uncertainties in the hydrauic mode and the measurement data. The procedure is appied to a sampe network for the case of a singe and of mutipe eakages and the effectiveness of the new methodoogy is demonstrated. The ocation of the eakage is correcty identified and its severity is accuratey computed, when the mode and measurement errors do not exceed certain threshod vaues above which the diagnosis of the system is not possibe. These threshod vaues depend on the configuration and the characteristics of the water distribution system under investigation and the ocation and severity of eakage as we as the number, ocation and type of sensors. More important, the present approach is capabe of identifying these threshod vaues beyond which no reiabe diagnosis is possibe. The ocation of the measuring devices has a significant effect on the reiabiity of the system identification. Optima sensor ocation strategies [23] can be used to improve the reiabiity of the eakage prediction estimates. The present approach may be part of an integrated software to assist decision-making for overa water network management strategy based on computer-aided toos. The main principes of this work can be extended to compressibe fuids. Acknowedgements This work has been partiay supported by the Greek Secretariat of Research and Technoogy (PENED ED580 and PAVET BE72) and by the Athens Water and Sewerage Company. This support is gratefuy acknowedged. References [1] Savic D, Waters G. Hydroinformatics technoogy and maintenance of UK water networks. J Qua Maintenance 1997;3(4): [2] Shamir U, Howard CD. An anaytic approach to scheduing pipe repacement. J Am Water Works Assoc 1979;71: [3] Ketter AJ, Gouter IC. An anaysis of pipe breakage in urban water distribution networks. Can J Civi Engng 1985;12(2): [4] Woodburn J, Lansey K, Mays LW. Mode for the optima rehabiitation and repacement of water distribution system components. Proc Nat Conf Hydrau Engng, New York 1987;

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