Entropy-Based Sensor Placement Optimization for ...

Water Resour Manage (2013) 27:4443–4468 DOI 10.1007/s11269-013-0419-8

Entropy-Based Sensor Placement Optimization for Waterloss Detection in Water Distribution Networks Symeon E. Christodoulou · Anastasis Gagatsis · Savvas Xanthos · Sofia Kranioti · Agathoklis Agathokleous · Michalis Fragiadakis

Received: 12 November 2012 / Accepted: 6 August 2013 / Published online: 5 September 2013 © Springer Science+Business Media Dordrecht 2013

Abstract The work presented herein addresses the problem of sensor placement optimization in urban water distribution networks by use of an entropy-based approach, for the purpose of efficient and economically viable waterloss incident detection. The proposed method is applicable to longitudinal rather than spatial sensing, thus to devices such as acoustic, pressure, or flow sensors acting on pipe segments. The method utilizes the maximality, subadditivity and equivocation properties of entropy, coupled with a statistical definition of the probability of sensing within a pipe segment, to assign an entropy metric to each pipe segment and subsequently optimize the location of sensors in the network based on maximizing the total entropy in the network. The method proposed is a greedy-search heuristic. Keywords Water distribution networks · Sensor placement optimization · Waterloss detection · Entropy

1 Introduction Each year, hundreds of kilometers of pipes across the globe are upgraded, or replaced, in an attempt to mitigate the effects of pipe bursts and water loss, and to maintain the uninterrupted transport of water. Existing water distribution systems are increasingly at risk due to numerous factors (both internal and external to the distribution networks) and the accidental or deterioration-based breakage of water distribution networks (WDN) represents a range of problems. S. E. Christodoulou (B) · A. Agathokleous Department of Civil and Environmental Engineering, University of Cyprus, 1678 Nicosia, Cyprus e-mail: [email protected] S. E. Christodoulou · A. Gagatsis · S. Xanthos · S. Kranioti · A. Agathokleous · M. Fragiadakis Nireas International Water Research Center, 1678 Nicosia, Cyprus

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S.E. Christodoulou et al.

Research to-date has helped identify a number of potential time-invariant and time-dependent risk factors contributing to pipe breaks. Among them, factors such as a pipe’s age, diameter, material and number of previous breaks, as well as the network’s operating pressure and water flow (Christodoulou and Deligianni 2010), soil aggressiveness, temperature, neighboring constructions, external loads (Kanakoudis 2004a), etc. The studies and methods reported upon in literature with regards to infrastructure assessment are primarily on deterioration modelling, pipe-break foreasting and network monitoring. The intent has been to assist owners of water distribution networks in improving their understanding of the systems’ behavior over time, their deterioration rate and their reliability with respect to presumed risk factors, so that they could intelligently arrive at “repair-or-replace” decisions on a more scientific basis. In terms of infrastructure assessment, the studies usually attempt to identify statistical relationships between water-main break rates and influential risk factors. Most such studies show a relationship between failure rates and the time of failure (age of pipes), and some of them suggest a method to optimize the replacement time of pipes (e.g. Kanakoudis and Tolikas 2001). Shamir and Howard (1979), for example, reported an exponential relationship and Clark et al. (1982) developed a linear multivariate equation to characterize the time from pipe installation to the first break and a multivariate exponential equation to determine the breakage rate after the first break. A study by Andreou et al. (1987) suggested a probabilistic approach consisting of a proportional hazards model to predict failure at an early age, and a Poisson-type model for the later stages, and further asserted that stratification of data (based on specific parameters) would increase the accuracy of the model. A non-homogeneous Poisson distribution model was later proposed by Goulter and Kazemi (1988) to predict the probability of subsequent breaks given that at least one break had already occurred. Finally, Kleiner and Rajani (1999) developed a framework to assess future rehabilitation needs using limited and incomplete data on pipe conditions. More recent studies on WDNs extended the studies on reliability with additional risk factors and analyses tools. Prasad et al. (2003) outlined a multi-objective genetic algorithm method and introduced a new reliability measure, called network resilience, that tried to provide surplus head above the minimum allowable head at nodes and reliable loops with practicable pipe diameters. Kanakoudis and Tolikas (2004) presented a method that hierarchically analyses the possible preventive maintenance actions in a water system, based on indices that assess the performance level of the system prior to any action taken, and accomplished through a technicoeconomical analysis related to failing system components. A demonstration of this method to the water supply system of the city of Athens, backing the method’s applicability, was subsequently presented by Kanakoudis (2004b). A number of frequently-used methods for the risk analyses of water supply systems were presented by Tuhovcak et al. (2006). The methods address the identification of qualitative and quantitative risks posed by the individual system components, the evaluation methods and interpretation of results, with an emphasis on the Hazard Analysis and the Critical Control Points (HACCP) method. Park (2008) presented a method to assess and track changes in the hazard functions between water main breaks by using the proportional hazards model.

Entropy-Based Sensor Placement Optimization

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The method provides a systematic framework for analyzing changes of the hazard functions as more breaks occur, and for identifying a critical point at which the hazard function changes to a different functional form. Finally, several findings on risk assessment and prioritization of ‘repair-or-replace’ actions were reported by Christodoulou et al. (2009, 2010), Christodoulou and Deligianni (2010), Christodoulou (2011) and Christodoulou and Agathokleous (2012) based on neurofuzzy systems, survival analysis and geospatial clustering of WDNs under both normal and abnormal operating conditions. Their findings reinforced the need for real-time monitoring of WDNs; a task which is among the most important and difficult ones for sustainably managing urban water distribution networks. Similar findings have also been reported by, among others, Kanakoudis and Tsitsifli (2011) and Tsitsifli et al. (2011) by use of methods such as survival analysis and the Discriminant Analysis and Classification (DAC) method. Real-time monitoring is nowadays essentially performed by supervisory control and data acquisition systems (SCADA) and by use of sensors strategically located across the network. The goal in placing the sensors is to maximize their sensing effectiveness while also limiting their deployment and operational cost, and within such a framework of real-time sustainable management of WDNs and waterloss detection, the issue of sensor-placement optimization is of high significance.

2 Sensor Placement—State of Knowledge As aforementioned, sensor placement optimization is a task critical to the efficient and sustainable management of WDNs and waterloss detection. Unlike monitoring spatial phenomena though, such as temperature, humidity, noise, or polution in an indoor or outdoor environment where sensors act radially, sensing in piping networks is restricted to longitudinal actions. In the case of spatial sensing, one approach is to assume that sensors have a fixed sensing radius and then to solve the task graphically or with GIS (Agathokleous et al. 2011), or as an instance of the art-gallery problem (Gonzalez-Banos and Latombe 2001). This would, for example, be the case of sensing soil moisture at ground level and associating abnormally high soil moisture with water loss. This geometric assumption of radially-fixed sensing, though, is inaccurate since, in reality, sensors make noisy and uneven measurements about the nearby environment and thus their sensing capability is not radially-invariant. Furthermore, signal correlations are not always characterized by radial geometries especially when more than one sensor is needed to localize a signal. An example of a spatially-based solution is reported by Agathokleous et al. (2011) where placement optimization of pressure and acoustic sensors in an urban WDN is addressed by means of a mathematical model coupled with a GIS module. The former optimization (pressure sensors) utilizes ground elevation contours so that, for every network’s sub-DMA, sensors are positioned in selected locations to fully monitor (and control) the pressure distribution across the network. The latter optimization (acoustic sensors) takes into account the sensor’s range as well as the number and location of the junctions in the water distribution network. The intent is to use network junctions as checkpoints for noise variations and to use graph theory and GIS-based spatial analysis for optimizing the location of such acoustic

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Fig. 1 Possible placement locations of acoustic sensors

sensors in the network. A sample analysis and snapshots of the produced output are shown in Figs. 1 and 2, with sensors of assumed distance coverage of 200m and with a desired level of redundancy of 2 sensors per location. Figure 1 shows the initial sensor locations (one at every street junction) and Fig. 2 shows the reduced number of sensors (optimized scenario) based on the desired level of redundancy and sensor sensitivity. The number of sensors reduces from 171 in the original topology (Fig. 1) to 79 in the revised topology (Fig. 2), thus representing a 54 % drop in the number of sensors utilized. Another example of a spatially-driven solution is by use of Voronoi diagrams (also known as Dirichlet tessellations). A Voronoi diagram is a special kind of decomposition of a given space, resulting in the partitioning of a plane with n points into convex polygons such that each polygon contains exactly one generating point

Fig. 2 GIS-derived placement locations of acoustic sensors


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Fig. 3 Area coverage based on Voronoi diagrams

and every point in a given polygon is closer to its generating point than to any other. Voronoi diagrams are used in partitioning an area into smaller areas around points of interest (e.g. nodes or sensors) so that the points within each subarea are equidistant from the sensors (Fig. 3). An alternative approach to spatial statistics (Cressie 1991; Caselton and Zidek 1984) is to treat sensing not as a spatial field but rather as a probabilistic data field following a multivariate normal distribution. The assumed probability distribution field is then used alongside a pilot deployment (or data from expert knowledge) to develop a Gaussian process (GP) model for the phenomena being sensed (Fig. 4).

Fig. 4 Sensor placement based on Gaussian processes (bivariate normal distributions)

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If, for example, the parameter being sensed is the operating pressure in the network then a simple approach would be to assume that the pressures have a (multivariate) Gaussian joint distribution (Deshpande et al. 2004) which can be developed and solved numerically using the method of Parzen windows (Parzen 1961). In the GPmodel approach, data from a pilot study or expert knowledge is used to learn the parameters of the underlying GP distribution and then the learned GP model is used to forecast data on the same distribution (Christodoulou 2004) or the effect of placing sensors at particular locations; thus optimizing their positions (Guestrin et al. 2005; Krause et al. 2008a, b). Caselton and Zidek (1984) proposed an alternative optimization criterion based on ‘mutual information’ which seeks sensor placements that are most informative about unsensed locations. The same criterion, coupled with a combinatorial optimization problem, was subsequently used by Krause et al. (2008a, b). Their algorithm combinatorially selects k out of n possible sensor locations by first utilizing a lazy evaluation technique that exploits submodularity to reduce significantly the number of sensor locations that need to be checked, and then reducing the order of computational complexity to O(kn) by exploiting locality in sensing areas. by trimming low covariance entries. They furthermore showed, how the submodularity of mutual information can be used to derive tight online bounds on the solutions obtained by any algorithm and then used submodularity to formulate a mixed-integer programming approach to compute the optimal solution using Branch and Bound techniques. Finally, they showed how mutual information can be made robust against node failures and model uncertainty, and how submodularity can again be exploited in these settings. As Krause et al. (2008a, b) report, many criteria have been proposed for characterizing the quality of placements given a GP model, including placing sensors at the points of highest entropy (variance) in the GP model. A typical sensor placement technique is to greedily add sensors where uncertainty about the phenomena is highest, that is, the highest entropy location of the GP (Cressie 1991; Shewry and Wynn 1987). A similar approach related to entropy and mutual information also reported by Guestrin et al. (2005) who suggested that sensors should be placed so as to maximize mutual information, and used a greedy-variance heuristic as an approximation to the problem. Their approach was to find the set of sensor locations that has maximum joint entropy, and in order to address the problem of sensors being placed far apart along the boundary and information being ‘wasted’, they proposed a weighting heuristic. Unfortunately, though, as Krause et al. (2008a, b) and Guestrin et al. (2005) report, “this criterion suf fers from a signif icant f law: entropy is an indirect criterion, not considering the prediction quality of the selected placements. The highest entropy set, that is, the sensors that are most uncertain about each other’s measurements, is usually characterized by sensor locations that are as far as possible from each other. Thus, the entropy criterion tends to place sensors along the borders of the area of interest (Ramakrishnan et al. 2005)... Since a sensor usually provides information about the area around it, a sensor on the boundary ‘wastes’ sensed information”. Entropy-related work on sensor placement optimization was presented by several researchers. GunHui et al. (2009) investigated the problem of determining optimal pressure monitoring locations and proposed a method based on entropy, defining entropy as the amount of information calculated from the pressure change due to


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the variation of discharge. When abnormal conditions occur at a node, the effect on the entire network was appraised and the actual pressure change pattern was calculated in EPANET. The optimal locations for pressure sensors was deemed to be the nodes having the maximum information from other nodes. Yang et al. (2008, 2010) presented a feature extraction and leak detection system using approximate entropy to discriminate the leak signal from the non-leak acoustic sources. Dorini et al. (2006, 2010), in response to a call for papers for the Battle of the Water Sensors Networks (BWSN), presented an optimal sensor placement methodology to assist in the effective and efficient detection of accidental and/or intentional contaminant intrusion(s), which was formulated and solved as a constrained multiobjective optimization problem. The solution methodology proposed was based on the Noisy Cross-Entropy Sensor Locator (nCESL) algorithm. The same call (BWSN) also led to the development and comparison of a number of methodologies, optimization models amd solution algorithms for locating sensors, as proposed by several researchers (Ostfeld et al. 2008; Krause et al. 2008a, b; Eliades and Polycarpou 2010; Eliades et al. 2011). Boulos (2007) presented an extension of the geospatial hierarchical selection approach implemented in the USEPA PipelineNET model for solving the sensor placement optimization problem for a wide range of practical monitoring objectives. The proposed approach can explicitly consider any combination of modeling scenarios, steady and unsteady flow conditions, multiple GIS layers, sensor placement costs, as well as a minimal distance between the sensors. Isovitsch and VanBriesen (2008) analyzed a water distribution system for optimal sensor placement based on four different intrusion scenarios and five different optimization criteria cases, and the spatial distributions of the selected sensor networks were analyzed and compared using a geographic information system to determine spatial trends in sensor placement. The relationship between sensor location and water demand was also analyzed using a geographic information system and a chisquare analysis. They concluded that sensor locations selected by minimizing the volume of consumed contaminated water or minimizing the population affected are likely to coincide with network nodes with a high reachable average demand. Alternatively, sensor locations selected by maximizing the detection likelihood are likely to coincide with network nodes with a low reachable average demand. Aral et al. (2010) provided a methodology for the optimal design of water sensor placement in water distribution networks, based on a simulation-optimization and a single-objective function approach which incorporates multiple factors used in the design of the system. Their proposed model mimics a multiobjective approach and yields the final design without explicitly specifying a preference among the multiple objectives of the problem, also introducing a reliability constraint. A progressive genetic algorithm approach is used for the solution of the model. Genetic algorithms were also the focus of Preis and Ostfeld (2008) who presented a modified genetic algorithm scheme for contaminant source characterization using three types of perfect and imperfect sensors: (1) perfect sensors providing accurate, unbiased, contamination concentration measurements; (2) sensors transmitting fuzzy measured information (i.e., high, medium, and low contamination); and (3) 01 (Boolean) sensors indicating only a contamination presence. Koch and McKenna (2011) proposed an approach for combining data from multiple stations to reduce false background alarms, by considering the location and

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time of individual detections as points resulting from a random space-time point process and by using Kulldorffs scan test to find statistically significant clusters of detections. Using EPANET to simulate contaminant plumes of varying sizes moving through a water network with varying amounts of sensing nodes, it was shown that the scan test could detect significant clusters of events, reduce the false alarms resulting from background noise and help indicate the time and source location of the contaminant. Hart et al. (2008, 2010) discussed their experience in developing and deploying both exact and heuristic algorithms for placing sensors in water distribution networks to mitigate against damage due to intentional or accidental introduction of contaminants, and limited-memory techniques that can optimize large-scale sensor placement problems. Perez et al. (2009) proposed a leakage detection method based on detecting significant discrepancies between pressure measurements and their estimations as obtained from the simulation of a calibrated water distribution network model. The proposed method uses the pressure sensitivity matrix to detect leakages, and identification of the best locations for sensor deployment is treated as an inverse problem which is solved using optimisation and genetic algorithms. Diwold et al. (2010) presented a population-based ant colony optimization algorithm called ‘WSP-PACO’ for sensor placement in water networks and tested its performance on two realistic water networks under several test conditions. These solutions were compared to solutions of previous studies on these networks, suggesting that the WSP-PACO algorithm is highly suitable for solving the sensor placement problem in water networks. Ant colony optimization was also the research subject of Afshar and Marino (2012) who presented a numerical procedure for the optimization of the position of water quality monitoring stations in a pressurized water distribution system (WDS). The procedure, which is based on the choice of the set of sampling stations which maximizes the monitored volume of water while keeping the number of stations at minimum, is formulated in terms of integer programming and the solution of the mathematical problem is efficiently approximated by means of a multi-objective multi-colony ant algorithm. A number of mathematical programming approaches are also presented in literature. Berry et al. (2003, 2006) present a mixed-integer programming formulation for sensor placement optimization in municipal water distribution systems that includes the temporal characteristics of contamination events and their impacts. The information is utilized in computing the impact of a contamination event over time and determining affected locations, by quantifying the benefits of sensing contamination at different junctions in the network. Berger-Wolf et al. (2005) considered two variants of sensor placement for contamination detection (sensor-constrained and time-constrained optimization), and showed that the sensor and time constrained versions of the problem are polynomially equivalent. Carr et al. (2006) presented a series of related robust optimization models for placing sensors in municipal water networks to detect contaminants that are maliciously or accidentally injected. The sensor placement problem is formulated as a mixed-integer programming problem, for which the objective coefficients are not known with certainty. They then consider a restricted absolute robustness criteron that is motivated by natural restrictions on the uncertain data, and define three robust optimization models that differ in how the coefficients in the objective vary. Watson et al. (2004, 2006) also presented


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mixed-integer linear programming models for the sensor placement problem over a range of design objectives, and by use of two real-world water systems, they showed that optimal solutions with respect to one design objective are typically highly sub-optimal with respect to other design objectives. The implication is that robust algorithms for the sensor placement problem must carefully and simultaneously consider multiple, disparate design objectives.

3 Sensor Placement Optimization and Entropy Maximization 3.1 Entropy: Overview Entropy, in physics, is a measure of the unavailability of a system’s energy to do work, and it is central to the second law of thermodynamics which deals with physical processes and the degree of spontaneity in their occurrence (spontaneous changes occur with an increase in entropy). Entropy is thus a measure of the smoothness with which a transformation occurs between the different states of a system. By extent, entropy has often been associated with the level of order/disorder in a thermodynamic system and it is a measure of the disorder and the amount of wasted energy during the transformation from one state to another. Within this context, the entropy relations derived by Landsberg (1984) provide the means for expressing the total amount of disorder (S D ) and order (S O ) in the system by use of: S D = C D /C I

(3.1)

S O = 1 − C O /C I

(3.2)

in which C D is the disorder capacity of the system, C I is the information capacity and C O is the order capacity of the system. 3.2 Entropy: Subadditivity and Maximality Properties Entropy (Hx ) in its classical definition is considered to be a metric of a system’s order and stability, and mathematically can be evaluated as the product of the probability mass function ( px ) of a variable x, times the natural logarithm of the inverse of the probability (Eq. 3.3). Hx =

x

px ln

1 px

(3.3)

Among the principal properties of entropy, two are of particular importance: subadditivity and maximality. Subadditivity, in mathematics, is a function’s property stating that the function’s value for the sum of two elements is always less than or equal to the sum of the function’s values for each element (Eq. 3.4). ∀x, y ∈ A, f (x + y) ≤ f (x) + f (y)

(3.4)

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In entropy terms, if a system consists of two subdomains having n and m components respectively then the total system entropy is less than or equal to the sum of the subdomains’ entropy (Eq. 3.5). H(v11, v12 , . . . , v1m , v21 , . . . , v2m , . . . , vn1 , . . . , vnm ) ⎞ ⎛ n n n m m m vi1 , vi2 , . . . , vim + H ⎝ v1 j , v2 j , . . . , vnj ⎠ (3.5) ≤H i=1

i=1

i=1

j=1

j=1

j=1

The maximality property states that the entropy function, H( p1, p2 , . . . , pn ), takes the greatest value when all admissible outcomes have equal probabilities. In other words, maximal uncertainty is reached for the equiprobability distribution of possible outcomes (Eq. 3.6). H( p1, p2 , . . . , pn ) ≤ H(1/n, 1/n, . . . , 1/n) iff pi = 1/n, ∀i = 1, 2, . . . , n

(3.6)

3.3 Entropy: Equivocation Property Equivocation is in effect the conditional entropy of one random variable against another, and it quantifies the remaining entropy (i.e. the uncertainty) of a random variable Y given that the value of another random variable X is known. In mathematical terms, equivocation is referred to as the entropy of Y conditional on X, is written as H(Y|X) and can be shown to be governed by the following equation: H(Y|X) ≡ p(x)H(Y|X = x) x∈X

=

p(x)

x∈X

=

x∈X,y∈Y

y∈Y

p(y|x) ln

1 p(y|x)

p(x) p(x, y) ln p(x, y)

(3.7)

It should be noted that the conditional entropy of Y given X, H(Y|X), is bound by the entropy of Y and that the joint entropy of Y and X, H(Y, X), is bound by the sum of the conditional entropies of H(Y|X) and H(X|Y). H(X|Y) ≤ H(X)

(3.8)

H(X, Y) = H(X|Y) + H(Y|X) + I(X, Y)

(3.9)

I(X, Y) ≤ H(X)

(3.10)

where I(X, Y) is the mutual information between X and Y. Equations 3.8–3.10 are showcased in Fig. 5, by use of a Venn diagram. For independent X and Y, H(Y|X) = H(Y) and H(X|Y) = H(X)

(3.11)

and for X governed by a uniform distribution H(X|Y = y) ≤ H(X) ∀X ∼ U(a, b )

(3.12)


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Fig. 5 Individual (H(X),H(Y)), joint (H(X,Y)), and conditional entropies for a pair of correlated subsystems X,Y with mutual information I(X,Y)

3.4 Entropy: Chain-Rule Property Based on the previous entropy properties and equations, the chain rule for conditional probability forms to be p(x) H(Y|X) = p(y|x) ln p(x, y) x∈X,y∈Y =−

p(x, y) ln p(x, y) +

x∈X,y∈Y

= H(X, Y) +

p(x, y) ln p(x)

x∈X,y∈Y

p(x) ln p(x)

x∈X

= H(X, Y) − H(X)

(3.13)

3.5 Entropy: Illustrative Example The subadditivity and maximality properties of the entropy function can be illustrated using Figs. 6 and 7. A hypothetical area requiring 8 sensors to be covered,

Fig. 6 Spatial illustration of entropy’s subadditivity and maximality properties

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Fig. 7 Numerical illustration of entropy’s subadditivity and maximality properties

may be planned in a number of possible ways. If no provision is made regarding the placement of the sensors, then one extreme would be to place all sensors very close to each other (Fig. 6a) without considering the total area coverage provided by them. This would result to a total entropy of HT = 0.000 (Fig. 7a). If consideration is given to the area covered by each sensor, then a number of different ways could be devised for placing the sensors, depending on how the sub-areas are used. For example, possible sensor placement combinations could be 1–7, 2–6, 3–5 or 4–4 (Fig. 6b–e). In the first case (Fig. 6b), HT = (−1/8) ln(1/8) + (−7/8) ln(7/8) = 0.377. For the other three cases (Fig. 6c–e), the corresponding total entropy values would be HT = 0.562, HT = 0.661 and HT = 0.693 respectively (Fig. 7c–e). Entropy values increase as more ‘pieces’ are created (subadditivity property) for a maximum value of HT = 0.693 when the distribution of sensors is uniform (maximality property). If it is further assumed that the sensors are distributed over four sub-areas (Fig. 6f and g), then the possible assignments of 1-1-3-3 and 2-2-2-2 sensors result in entropy values of HT = 1.256 and HT = 1.386 respectively (Fig. 7f and g). Again, entropy increases as the number of subdivisions increases, maximizing at the equiprobability distribution. Similarly, as we go to a 6-subdivision or an 8-subdivision area coverage the subadditivity and maximality property still hold, resulting in maximal entropy values of HT = 1.733 and HT = 2.197 respectively (Fig. 7h–j). The maximal value (Fig. 7j) corresponds to the case of an 8-subdivision sensor placement area coverage, with one sensor placed in each of the 8 subdivisions (Fig. 6j). This is the case of an equiprobable distribution of the sensors. It should also be noted that, should one consider the area coverage achieved with each sensor arrangement then the entropy also increases as the distribution flattens (equiprobability) and as the coverage increases (more subdivisions are covered). For example, for 2 subdivisions (area coverage equals 2/8 = 0.25) the highest entropy is 0.693 (Figs. 6e and 7e). For 4 subdivisions (area coverage equals (4/8 = 0.50) the highest entropy is 1.386 (Figs. 6g and 7g).


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Even though the concept of entropy has also been introduced to disciplines other than physics and chemistry, it has yet to be systematically applied in water resources management and sensor placement optimization. 3.6 Restatement of the Sensor Placement Optimization Problem Using Entropy Maximization Since, according to the concept of entropy, entropy is a good measure of a system’s order and stability, maximizing in value when a system is at an ‘equiprobability’ state, then a higher degree of entropy should also indicate a more-balanced system; one in which the information generated and/or distributed among its parts are of equal value. The sensor-placement optimization problem can thus be restated as an entropy-maximization problem: how many sensors are needed and in what locations should be placed so that the information/knowledge acquired is maximized, by use of sensor sources of equivalent information value? Within the aforementioned problem restatement, the stochastic definition of probability px can be substituted with a statistical definition, replacing the value of px in the entropy equation (Eq. 3.3) with the ratio of a sensor’s sensing radius over the total length of the network. The total network entropy (HT ) for a single-type sensor would then become nt nt ri ri 1 ri HT = =− (3.14) ln ln LT ri /LT LT LT i=1 i=1 where ri is the sensing radius of sensor i, nt is the total number of sensors in the network, and LT is the total length of the network. The equation, though, is based on a factor (r/LT ) that is not exhibiting classical probability properties when the placed sensors do not cover the entire network length, and thus is not mathematically correct. The above equation can be further adjusted in order to conform to classical probability and entropy theory properties, to account for the fact that sensors at junctions of multiple arcs contribute to the entropy levels of these arcs, and to avoid clustering of sensors at only a few parts of the network, by taking the ratio of r/L based on the arc length and not the network length. Thus, the value of px is taken to be the ratio of the sensor’s radius over the length of the network arc being sensed, and the total system entropy can be computed by summing up the entropy values for each arc. nt nt ri ri 1 ri HT = =− (3.15) ln ln L r /L L L i i i i i i=1 i=1 For multiple sensor types, the total network entropy can, similarly, be defined as HT = −

nt nr ri, j j=1 i=1

r T, j

ln

ri, j r T, j

(3.16)

where j is the sensor-type index; nr is the number of different sensor types used in the project; ri, j is the number of units of sensor type j used on node i; nt is the total

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number of sensors in the network and r T, j is the total number of units of sensor type j used in the network. The goal, therefore, when optimizing sensor placement within the aforementioned entropy framework, is to maximize the network’s total entropy, subject to the imposed sensor/resource constraints. This optimization is given by ⎫ ⎧ ⎬ nt nr ⎨ ri, j ri, j max(HT ) = max − (3.17a) ln ⎩ r T, j r T, j ⎭ j=1 i=1

subject to:

nt ri, j ≤ r T, j ∀ j

(3.17b)

i=1

ri, j : integer ∀ j

(3.17c)

ri, j > 0

(3.17d)

i, j : integer

(3.17e)

The constraints in the above equations refer to, respectively, (i) the overall system sensor-availability constraints, (ii) the assumption that sensor assignments are of integer value (fractional sensor assignments are disallowed as non-physical assignments), and (iii) the exclusion of zero-value sensor assignments for any of the network nodes requiring sensors.

4 Entropy and Urban Water Distribution Networks 4.1 Proposed Method As aforementioned, the goal is to maximize the network entropy, subject to an allowable maximum number of sensors or, equivalently, to maximize the entropy while minimizing the number of sensors used. Unlike terrestrial sensors or antennas, though, sensors on urban water distribution networks act along the pipe segments and their coverage is in effect their sensing length acting longitudinally along the length of the pipes being monitored. Thus, in the proposed entropy-maximization process, the entropy levels are evaluated at the arc level (i.e. the pipe segment level) and not at the network level. This makes the proposed entropy equation more suitable for piping networks, mathematically more robust and avoids the clustering of sensors in small parts of the network. Furthermore, in order to account for the overlap in sensing radii of sensors

Fig. 8 Proposed entropy-based sensor placement method: notation used


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Fig. 9 Proposed entropy-based sensor placement method: generic cases of arc length a smaller and b greater than the sensing radii of sensors at the end-nodes

placed at the end-nodes of an arc and/or segment lengths shorter than the sensor’s sensing radius, the value of ri used in Eq. 3.15 is taken as the minimum between the segment length, Li , and the sensor’s sensing radius, xi (in the case of one sensor), or the combined sensing radii (in the case of two sensors). ri = min {xi ; Li }

(4.1)

Equation 4.1 can be visually demonstrated by use of Figs. 8, 9, 10, 11, 12 and 13. Figure 8 depicts the notation used in developing the proposed sensor placement method. A filled circle at node ni indicates that a sensor is placed at this node, while an empty circle indicates that no sensor is placed at the node (e.g. Fig. 8, node n j ). For the pipe pictured in Fig. 8 the total pipe entropy can be computed as [−x/L ∗ ln(x/L)]. In Fig. 9a, the pipe segment in examination is longer than the sum of the sensing radii of the two nodal sensors used. In this case, the total pipe entropy can be computed as [− min (2x; L) /L ∗ ln {(2x; L) /L} = −2x ∗ ln(2x/L)]. Figure 9b, depicts the case where the pipe segment in examination is shorter in length than the sum of the sensing radii of the two nodal sensors used. In this case, the total pipe entropy is taken as [− min (2x; L) /L ∗ ln {(2x; L) /L} = −L ∗ ln(L/L) = 0]. For such pipes, of which the length is smaller than the sum of the sensing radii of two nodal sensors (one at each pipe end), the entropy approach results in zero entropy values. If only one sensor is used (at either of the nodes), the entropy value is higher than zero, thus the entropy-maximization approach gives higher preference to a singe-node arrangement compared to the two-node arrangement. A numeric demonstration can be seen in Fig. 10. Should one sensor be used (at node ni ) then the entropy for pipe (ni , n j ) is computed to be [−200/300 ∗ ln(200/300) = 0.270]. If two sensors are used (at nodes ni and n j ) then the entropy is computed to be [−300/300 ∗ ln(300/300) = 0.000].

Fig. 10 Proposed entropy-based sensor placement method: numeric example for the case of sensors at a one end-node and b both end-nodes, for an arc with length between 1x and 2x the sensor’s sensing radius

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Fig. 11 Proposed entropy-based sensor placement method: numeric example for the case of sensors at a one end-node and b both end-nodes, for an arc with length equal to the sensor’s sensing radius

The same preference for a single sensor compared to sensors at both end-nodes is also manifested when the pipe length is equal to the sensing radius. For example, as shown in Fig. 11a, the entropy for pipe (ni , n j) when only one sensor is used is computed to be [−200/200 ∗ ln(200/200) = 0.000]. If two sensors are used (at nodes ni and n j ) then the entropy is again computed to be [−200/200 ∗ ln(200/200) = 0.000]. Since both arrangements produced the same entropy level, the preferred arrangement is the one that utilizes a smaller number of sensors, i.e. Fig. 11a. The proposed entropy-maximization approach holds true in more complicated sensor arrangements as well. For example, when a three-segment pipe junction is investigated (Figs. 12 and 13), the method concludes that a signle-node sensor at the junction of the pipes is better than sensors at the outer nodes. In Fig. 12, both sensor arrangements produce network entropy values of [HT = 3 ∗ (−200/200) ∗ ln(200/200) = 0.000], but the sensor arrangement depicted in Fig. 12a uses fewer sensors and thus is preferred. Similarly, in Fig. 13, both sensor arrangements produce network entropy values of [HT = 3 ∗ (−200/300) ∗ ln(200/300) = 0.811], but the sensor arrangement depicted in Fig. 13a uses fewer sensors and thus is preferred. 4.2 Case-Study Pipe Network Let us consider a more complicated piping network example, consisting of 17 nodes and 18 arcs (pipes), as shown in Fig. 14. The pipes vary in length and nodal

Fig. 12 Proposed entropy-based sensor placement method: numeric example for the case of sensors at a the junction of three arcs and b at the ends of a three-arc junction, for arcs with length equal to the sensor’s sensing radius


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Fig. 13 Proposed entropy-based sensor placement method: numeric example for the case of sensors at a the junction of three arcs and b at the ends of a three-arc junction, for arcs with length between 1x and 2x the sensor’s sensing radius

connectivity, and the network nodes are initially assumed having no sensors (thus the original network entropy level is zero). The network entropy for the topology depicted in this case study is expressed by the following equation: HTSYSTEM = H1,2 + H2,3 + H2,6 + H3,4 + H4,8 + H4,5 + H5,6 + H6,7 + H5,10 + H8,9 + H10,11 + H9,10 + H9,16 + H9,17 + H11,13 + H12,13 + H13,14 + H11,15

Fig. 14 Case study—network topology; sensors at all nodes

(4.2)

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The network entropy is then evaluated by use of Eq. 4.2, assuming that sensors are placed at all nodes (Fig. 14; Eq. 4.3). This constitutes the upper bound in terms of the number of sensors utilized and the corresponding entropy (HT = 2.089) is the base value for which improvement is sought. The sensors are assumed to have a sensing radius of 200 m.

200 200 300 300 400 400 ALL HT = − ∗ ln + − ∗ ln + − ∗ ln 200 200 300 300 600 600

400 400 400 400 400 400 ∗ ln + − ∗ ln + − ∗ ln + − 400 400 500 500 500 500

400 400 400 200 200 400 ∗ ln + − ∗ ln + − ∗ ln + − 400 400 800 800 200 200

400 400 200 200 300 300 + − ∗ ln + − ∗ ln + − ∗ ln 600 600 200 200 300 300

200 200 400 400 400 400 + − ∗ ln + − ∗ ln + − ∗ ln 200 200 700 700 500 500

300 300 400 400 400 400 + − ∗ ln + − ∗ ln + − ∗ ln 300 300 400 400 800 800 = 2.089

(4.3)

As a further investigation of the entropy-maximization method and a prelude to the proposed algorithm, consider three additional sensor arrangements: (1) sensors at even-numbered nodes, (2) sensors at odd-numbered nodes, (3) sensors clustered at one side of the network (nodes 1–8). In the first case (Fig. 15; HTEVEN ) the network

Fig. 15 Case study—sensors at even-numbered nodes


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entropy evaluates to 3.662, in the second case (Fig. 16; HTODD ) the network entropy evaluates to 4.001 and in the third case (HT1,2,3,4,5,6,7,8 ) the network entropy evaluates to 1.161. Let us now consider the entropy-maximization approach. The method starts with the nodal entropy values from the full-sensored stage (Fig. 14) as the calculation base and assumes that the entropy contributions to the total network entropy from the nodal sensors are not subject to the equivocation property and Eq. 3.15. Upon ranking the nodal entropies in descending order, the method selects the node that contributes the maximum to the network entropy and places a sensor at that node. For the case-study network, the node to first receive a sensor is node ‘4’ (as shown in Table 1). Upon assigning a sensor at a node, the entropy values of the connecting nodes are adjusted, considering equivocation and the entropy-maximization approach (Eq. 3.15 and 4.1). For example, having placed a sensor at node ‘4’ (thus monitoring pipes ‘4,5’, ‘4,3’ and ‘4,8’), the entropy values of nodes ‘5’, ‘3’ and ‘8’ need to be adjusted so that the equivocation property applies to pipes ‘5,4’, ‘3,4’ and ‘8,4’. In such case, the total entropy for pipe ‘5,4’ becomes [−400/500 ∗ ln(400/500) = 0.179] and thus, given that the entropy contributed by node ‘4’ is 0.367, the entropy of node ‘5’ should be adjusted from 0.367 to −0.188. Similarly, the entropy values for nodes ‘3’ and ‘8’ along pipes ‘3,4’ and ‘8,4’ are adjusted to −0.347 and −0.188 respectively (Table 1). The revised nodal entropies are calculated, tabulated and sorted again, and the node with the highest entropy is selected for sensor placement. Node ‘6’ is the next node to receive a sensor (Table 1). The pipes affected are ‘6,7’, ‘6,2’ and ‘6,5’, with nodes ‘7’, ‘2’ and ‘5’ subject to the equivocation property and the proposed entropy-maximization approach (Eq. 3.15 and 4.1). The entropy adjustements at these nodes are calculated and tabulated (Table 1) leading to the identification of node ‘9’ as the third node to receive a sensor.

Fig. 16 Case study—sensors at odd-numbered nodes

14 15 16 17

12 13

11

10

9

7 8

6

5

4

3

Node 1 2

Arc 1,2 2,1 2,3 2,6 3,2 3,4 4,5 4,3 4,8 5,4 5,6 5,10 6,7 6,2 6,5 7,6 8,4 8,9 9,8 9,10 9,16 9,17 10,5 10,9 10,11 11,10 11,13 11,15 12,13 13,11 13,12 13,14 14,13 15,11 16,9 17,9

Length 200 200 300 600 300 400 500 400 500 500 400 200 800 600 400 800 500 600 600 300 200 700 200 300 200 200 500 800 300 500 300 400 400 800 200 700

H(X) 0.000 0.000 0.270 0.366 0.270 0.347 0.367 0.347 0.367 0.367 0.347 0.000 0.347 0.366 0.347 0.347 0.367 0.366 0.366 0.270 0.000 0.358 0.000 0.270 0.000 0.000 0.367 0.347 0.270 0.367 0.270 0.347 0.347 0.347 0.000 0.358

0.347 0.347 0.000 0.358

0.270 0.983

0.713

0.270

0.994

0.347 0.733

1.059

0.713

0.617

HT 0.000 0.637

Sensor 1

[H(Y|X)] – – – – – −0.347 – – – −0.188 – – – – – – −0.188 – – – – – – – – – – – – – – – – – – – 0.347 0.347 0.000 0.358

0.270 0.983

0.713

0.270

0.994

0.347 0.545

0.525

1.080

0.270

HT 0.000 0.637

Table 1 Entropy-based sensor placement optimization (case study, sensors 1–3) Sensor 2

[H(Y|X)] – – – −0.096 – – – – – – −0.347 – – – – – – – – – – – – – – – – – – – – – – – – – 0.347 0.347 0.000 0.000

0.270 0.983

0.713

0.270

0.347 0.545

1.059

0.179

1.080

0.270

HT 0.000 0.541

Sensor 3

[H(Y|X)] – – – – – – – – – – – – – – – – – −0.096 – – – – – −0.270 – – – – – – – – – – – −0.038

0.347 0.347 0.000 0.320

0.270 0.983

0.713

0.000

0.994

0.347 0.449

1.059

0.179

1.080

0.270

HT 0.000 0.541

4462 S.E. Christodoulou et al.

14 15 16 17

12 13

11

10

9

7 8

6

5

4

3

Node 1 2

Arc 1,2 2,1 2,3 2,6 3,2 3,4 4,5 4,3 4,8 5,4 5,6 5,10 6,7 6,2 6,5 7,6 8,4 8,9 9,8 9,10 9,16 9,17 10,5 10,9 10,11 11,10 11,13 11,15 12,13 13,11 13,12 13,14 14,13 15,11 16,9 17,9

Length 200 200 300 600 300 400 500 400 500 500 400 200 800 600 400 800 500 600 600 300 200 700 200 300 200 200 500 800 300 500 300 400 400 800 200 700

H(X) 0.000 0.000 0.270 0.366 0.270 0.347 0.367 0.347 0.367 0.367 0.347 0.000 0.347 0.366 0.347 0.347 0.367 0.366 0.366 0.270 0.000 0.358 – 0.270 – 0.000 0.367 0.347 0.270 0.367 0.270 0.347 0.347 0.347 0.000 0.358

Sensor 4

[H(Y|X)] – – – – – – – – – – – – – – – – – – – – – – – – – – −0.188 – −0.270 – – – −0.347 – – – 0.000 0.347 0.000 0.320

0.000 0.983

0.525

0.000

0.994

0.347 0.449

1.059

0.179

1.080

0.270

HT 0.000 0.541

Sensor 5

Table 2 Entropy-based sensor placement optimization (case study, sensors 4–6) [H(Y|X)] – – – – −0.270 – – – – – – – – −0.096 – – – – – – – – – – – – – – – – – – – – – – 0.000 0.347 0.000 0.320

0.000 0.983

0.525

0.000

0.994

0.347 0.449

0.963

0.179

1.080

0.000

HT 0.000 0.541

Sensor 6

[H(Y|X)] – – – – – – – – – – – – – – – – – – – – – – – – – – – – – −0.188 – – – – – – 0.000 0.347 0.000 0.320

0.000 0.795

0.525

0.000

0.994

0.347 0.449

0.963

0.179

1.080

0.000

HT 0.000 0.541

Sensor 7

Entropy-Based Sensor Placement Optimization 4463

4464 Table 3 Entropy values for various sensor topologies (case study)

S.E. Christodoulou et al. Number of sensors

Sensor configuration

Total entropy

1 2 3 4 5 6 7 8 8 9 17

4 4, 6 4, 6, 9 4, 6, 9, 13 4, 6, 9, 13, 2 4, 6, 9, 13, 2, 11 4, 6, 9, 13, 2, 11, 8 2, 4, 6, 8, 10, 12, 14, 16 1, 2, 3, 4, 5, 6, 7, 8 1, 3, 5, 7, 9, 11, 13, 15, 17 1–17

1.080 2.139 3.133 4.117 4.290 4.540 4.348 3.662 1.161 4.001 2.089

The process is repeated until the maximum allowable number of sensors is reached. If, in this case-study network, we assume that a constraint on the number of sensors (nt ) is imposed of nt = 6, then the proposed entropy-maximization approach arrives at the sensor topology shown in Fig. 16 and tabulated in Tables 1, 2 and 3. The total network entropy of the entropy-derived sensor topology (Fig. 17) can be calculated to HT = 4.540 by use of Eq. 4.2. As shown in Fig. 18, the entropy optimization heuristic reaches maximality not when all nodes are used as sensor positions, and not when sensors are placed at the boundary of the network. Furthermore, convergence to a solution is achieved in relatively few iterations of the greedy-search algorithm.

Fig. 17 Case study—sensors at entropy-derived nodes


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Fig. 18 Case study—total network entropy vs. sensor placement topologies

5 Conclusion The work presented herein addresses the problem of sensor placement optimization in water distribution networks by use of an entropy-based approach. The goal in optimizing sensor placement is twofold: (i) the sustainable and economically viable management of water distribution networks, and (ii) the efficient waterloss incident detection. The proposed method is a greedy-search heuristic that utilizes the maximality, subadditivity and equivocation properties of entropy to formulate the sensor placement optimization problem as an entropy-maximization problem, with entropy defined as a ratio of sensing length over pipe length and maximized at the network level. The proposed approach is applicable to longitudinal rather than spatial sensing (thus to devices such as acoustic, pressure, or flow sensors acting on pipe segments), is robust, is independent of the network’s operating parameters and shows a quick convergence to optimal or near-optimal solutions. Furthemore, as the case-study network shows, the proposed method does not yield solutions with sensor locations as far as possible from each other, or with sensors placed along the borders of the area of interest, as Ramakrishnan et al. (2005) noted in their work about entropy and mutual information. Future work on sensor placement and on the proposed entropy-maximization heuristic entails the following actions: –

–

Improve the proposed method and apply the algorithm to a large-scale urban water distribution network, linking the algorithm with GIS and spatio-temporal analysis. Expand the method to incorporate a network’s operating parameters further to its topology. This will allow the algorithm to consider real-time operating

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–

– –


pressure and flow data, as well as digital elevation models to better optimize sensor locations. Use real-time data acquisition and automatic meter reading (AMR) as a backbone of sensor placement optimization. The work on a new-generation intelligent AMR device is ongoing and pilot tests are soon to be contacted. Use sensor placement optimization to minimize the time-to-detect waterloss or contamination incidents in the piping network. Utilize the optimized sensor location to increase the network’s reliability against catastrophic events.

Acknowledgements The work presented herein is part of research initiatives of the NIREAS International Water Research Center, co-financed by the European Regional Development Fund and the Republic of Cyprus through the Cyprus Research Promotion Foundation (Grants No. NEA YPODOMI/STRAT/ 0308/09, AEIFORIA/ASTI/0609(BIE)/07 and PENEK/ENISH/0308/34).

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