Multi-objective Sensor Placements with Improved Water Quality Models in a Network with Multiple Junctions

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1 451 Multi-objective Sensor Placements with Improved Water Quality Models in a Network with Multiple Junctions R.G. Austin 1, C. Y. Choi 1, A. Preis 2, A. Ostfeld 3, and K. Lansey 4 1 Department of Agricultural and Biosystems Engineering The University of Arizona, Tucson, Arizona 85721, U.S.A. 2 Center for Environmental Sensing and Modeling, MIT-SMART Center, Singapore 3 Department of Civil Engineering and Environmental Engineering Technion, Israel Institute of Technology, Haifa 32000, Israel 4 Department of Civil and Engineering Mechanics The University of Arizona, Tucson, Arizona 85721, U.S.A. Concerns about the security of water distribution systems have lead to increased interest in sensor placement in water distribution systems. Due to the cost of both placing and maintaining these sensors, the number of sensors used must be limited. These constraints make the sensor deployment locations crucial in a water monitoring system. Many studies, based on differing algorithms and objective functions, have sought to determine ways to optimize sensor location. These studies have largely relied on current water quality models that assume perfect mixing at pipe junctions. However, it has been shown that using a water quality model that accounts for imperfect mixing () at pipe intersections produces outcomes that differ from those produced by studies that assume perfect mixing and, consequently produces a different scheme for optimal sensor placement. The current work uses a multiobjective approach that relies on the nondominated, sorted algorithm II. The study seeks, first, to contrast the use of the water-quality model to the use of water quality models that assume perfect mixing, and, second, to propose a more comprehensive approach to sensor placement. By using a simpler objective of optimizing for complete sensor coverage, the study will expand on pervious work that made this comparison. An example network is analyzed using both and, and the results are compared. 1. INTRODUCTION Sensor Placement in Water Distribution Systems The Safe Water Drinking Act, issued by the United States Environmental Protection Agency in 1990, requires water utilities to monitor water distribution systems (WDS) for water quality. Since the passing of this act, there has been increased interest in sampling and, in turn, determining the best sampling locations. Lee and Denininger (1992) first approached the problem using optimization tools to determine the best locations; their analysis was based on fractions of water contributed to monitoring locations of a water distribution network. Since that time, there have been many advances in methodologies for monitoring water quality

2 452 The terrorist events of September 11 th generated a greater demand for increasing the security of water distribution systems, and many subsequent studies, recognizing the possibility of intentional contamination, have focused on sensor placement. A primary goal in designing a sensor placement scheme should be to give the water distribution operator the best possible tool for making informed decisions about which locations to monitor. Ideal would be a sensor network that provides (1) rapid and (2) accurate detection of an event that takes place anywhere in the system. Obtaining such perfection is, however, simply impossible, given the current technology and budget constraints. Because continuous, ubiquitous monitoring is expensive, the number of sensors deployed must be kept to the minimum needed to achieve threshold levels of accuracy and timeliness as determined by the designer who wishes to make good use of limited resources and must simultaneously weigh the importance of several goals. Multiobjective Optimization In contrast to single objective schemes, a multiobjective optimization scheme accounts for several different objective functions simultaneously. Formally, this can be T written F ( x) = [ f1( x), f 2 ( x),..., f M ( x) ] where F(x) is the multiobjective function that is composed of f x), f ( x),..., f ( ) which are a set of objective functions, and 1( 2 M x T x = ( x1, x2,..., x n ) is the decision variable vector. The goal of this type of optimization is to find the set of all of the best solutions, subject to any constraints imposed by the problem. The best solutions are solutions such that no other solutions may be found that improve any of the objective functions without making another objective function value worse. This set of solutions is called the Pareto optimal solution set. Water Quality Model and Optimization Most WDS water quality modeling software relies on the assumption of perfect mixing at all junctions within the system. Van Bloemen Waanders et al. (2005) showed, through computational fluid dynamics (CFD), of a cross fitting that this assumption would cause significant errors. They determined that two incoming adjacent flows bifurcate rather than mix. Romero-Gomez et al. (2008a), in describing this mixing phenomenon over a large range of the Reynolds number using CFD and Austin et al. (2008), established an experimental database to model mixing behavior in the turbulent flow regime. Both computational and experimental results demonstrated that nodal mixing could be dramatically different from those produced by a perfect mixing model. Choi et al. (2008b) showed that mixing was mostly dependent on relative inflows and outflows rather than on Reynolds number, and that some modeling simplifications could be made in order to obtain a reasonable estimate of mixing over both the laminar and transitional flow regimes as well. Along with cross fittings, wye and tee junctions have been investigated and shown to exhibit imperfect mixing behavior. These include situations in which more than one tee or wye type fitting have been placed in close proximity. Choi et al. (2008b) developed a database for these other fitting configurations. The water quality model is a modified version of, one that implements an imperfect mixing model for cross, double tee, and tee and wye fittings. The program was created by a group of researchers at the University of Arizona (Choi et al., 2008a;, 2009) and based on data sets collected through experimental trials.

3 453 The program uses s standard hydraulic water quality solvers but alters the water quality code by calculating concentrations downstream from junctions based on imperfect mixing, when appropriate, and using an interpolation of data. Romero-Gomez et al. (2008b) demonstrated that could be used as the water quality model in the single objective optimization scheme to maximize system coverage (i.e., likelihood of detection). A minimum hazard level was used in this study to construct a pollution matrix with a binary variable to signal whether or not a contamination above the minimum hazard level had been detected. It was shown that, depending on the minimum hazard level, the water quality model (perfect mixing vs. ) had a significant influence on the number of sensors required for complete coverage and that, due to the increased accuracy of the water quality model, sensor designs based on should be more reliable than those based on perfect mixing. Here, we pose a multi-objective optimization problem to extend the comparison of perfect and imperfect mixing assumptions. 2. METHODS Optimization Functions The optimization problem is formulated using the three objective functions proposed by Preis and Ostfeld (2008): sensor detection likelihood, sensor detection redundancy, and sensor expected detection time. Detection Likelihood Detection likelihood is the probability that a random event in the pollution matrix will be detected at one of the sensor locations. This metric is intended to indicate the degree to which the sensor locations cover the network. A network with high detection likelihood has a good chance of detecting a random event, while an arrangement of sensors with low detection likelihood has a low detection probability. More formally, the sensor detection likelihood is given as: S 1 f1 = d r (1) S r= 1 where f 1 is the sensor detection likelihood, d r =1 if there is detection at least one of the sensor locations or d r =0 otherwise, and S is the total number of pollution events. The detection likelihood indicates the proportion of events in the randomly generated set that would be detected. As such, it is desirable to maximize its value. Detection Redundancy Detection redundancy is defined as the probability that if an event occurs, a redundant detection will also occur. Thus, sensor detection redundancy is intended to increase the reliability of the sensor network by reducing the number of false detections that can occur at a single sensor. Mathematically, the term is defined as: S 1 f 2 = S Rr (2) r 1 d = r r= 1

4 454 where f 2 is the sensor detection redundancy, and R r =1 if the redundancy condition is met and R r =0 otherwise. In our specific examples, three sensors are used as the number that must be triggered in order to have a redundant condition. Sensor detection redundancy is to be maximized. Detection Time Expected detection time is a metric defining the time that passes from the moment when an event occurs until it is detected. This metric thus specifies the expected time that a given sensor configuration should need in order to initially detect a contamination. The metric is calculated as: f = E( t ) (3) 3 d where f 3 is the sensor detection time, t d is the time from contamination until the first sensor detection, and E(t d ) is the expected value of t d. To mitigate harm caused by a contamination event, detection time is minimized in order to minimize the amount of time that passes before action can be taken. The sensor detection time is to be minimized in order to minimize the time that passes before action can be take to mitigate harm caused by a contamination event. Pollution Matrix Generation The purpose of the pollution matrix is to organize the results of a set of simulated pollution events into a form that can be easily evaluated by our objective functions (Ostfeld and Salomons, 2004). In this study, a pair of pollution matrices was created for each of the networks. One matrix was derived from a water quality analysis using, while the second other matrix was developed using as the simulation model. Aside from the difference of the water quality solver, each matrix was created using the same procedure that was automated through a C++ program using the toolbox. The program ran both hydraulic and water quality simulations for each pollution event. In the pollution matrix, each line contains the record of a contamination event with a corresponding injection node, mass rate, time, and duration. The pollution information follows the injection information and indicates the concentration of each contamination initially detected for each node being considered. The time of first detection, and the duration of the contamination at that node is also included. Separate pollution matrices were constructed for each minimum detection limit (MDL) that was set to represent which represented the pattern of detection based on the corresponding level where detection would occur. NSGA-II Optimization Algorithm The nondominated, sorted genetic algorithm developed by Deb et al. (2000) and implemented by Preis and Ostfeld (2008) was used as the optimization solver by modifying the function evaluation routine to incorporate as well as EPPANET. Briefly, the NSGA-II algorithm is a multiobjective genetic algorithm that generates a random set of strings to represent the initial population of sensor locations. Each string s fitness is then evaluated for all objective functions. The strings are then sorted and nondominated strings are identified. One string dominates another when:

5 Have a fitness value as good as the other string for all fitness functions 2. Have a fitness value better than the other string for at least one objective function. Nondominated strings are then sorted according to the average distance between the fitness vectors of the other strings, and solutions that favor greater distances (diversity in the solution) are selected. The algorithm then produces a new generation of strings based on selection, mutation, and a crossover based on the best set of available solutions from the previous generation. This process of evaluation, sorting, and new population generation is repeated until a stopping criteria is reached. In this study, the solutions were all run for 1,000 generations. Network Descriptions This exemplary network is part of the freeware package and consists of 120 links, 90 nodes, 3 elevated tanks, and 2 pumps that connect to 2 reservoirs (a lake and a river). Eight nodes from the network are configured as 4-way junctions (circled in Figure 1) and were thus modified in order to comply with the requirement of for such junctions. The modifications led to a combination of all the types shown by Choi et al. (2008b): two crosses, two double-tee, one YU-type, and three YN-type junctions. The location and characteristics of the new junctions are listed in Table 1. example Network3 is used as the application network in this study. The network was slightly altered so that it would better suite the study. These alterations follow those implemented by Romero-Gomez et al. (2009). Incomplete mixing occurs at nodes (a) having two incoming and two outgoing pipes or (b) that are a special connection. In original Nework3, only cross junctions (type (a)) are available. Therefore, changes were made in the network to introduce special junction types, which were modeled in (cross, double tee, etc.) and that both the entering and exiting pipes in these nodes where all the same size. The changes introduced 8 special junctions that would produce imperfect mixing: two crosses, two double-tees, one YU-type, and three YN-type junctions (for junction types, see Choi et al., 2009). To ensure that only pipes of the same size entered and exited the special junctions, extra nodes were added around each junction along with a short length of added pipe. These nodes were not used as injection or detection locations. 3. RESULTS The pollution matrices were created by including events for every possible combination of events from a list of injection mass rates, injection durations and start times. All of the possible event locations were used for each of these combinations. The pollution matrices for Network 3 was done with 2 mass rates of 50000, and mg/min, durations of 2 and 4 hrs, with 4 start times distributed evenly throughout the day. Sets of pollution matrices were created for each MDL (0, 0.1, 0.01, 0.001, and mg/l).

6 456 Figure 1. Network 3 from examples. Network was modified at the circled nodes that correspond to 4-way junctions. Table 1. Modifications on Network 3 example from the package (for modified junction types, see Choi et al., 2009). Node ID Type Junction diameter (D) Dimensionless distance (L/D) 121 YN N X YN N X YU YN

7 457 Incomplete mixing causes concentration changes downstream of a junction where mixing is incomplete. If additional junctions have incomplete mixing downstream, the difference between the plumes for perfect and imperfect mixing may amplify. The concentration differences are apparent when applying alternative a minimum detection levels (MDL). These differences can be seen between the Pareto fronts for the two mixing conditions. The Pareto fronts for detection likelihood and detection redundancy are shown in Fig. 2 for 4 MDLs. As the MDL is increased, contamination events are missed at more locations resulting in a shift on the Pareto front. Comparing the results from and we see that at most MDLs there is a slight difference between the two methods that diminishes as the MDL is decreased. Little difference can be seen at the mg/l level and with perfect sensors (an MDL of zero) the results for and were very similar and are not shown. At low MDLs, any contaminant is detected and a small concentration discharging from an imperfectly mixed node is equivalent to a uniformly mixed higher concentration and will be detected at the assumed precise meters. The differences between the two methods are especially prominent with points that are at the upper range of the detection likelihood parameter. In this portion of the Pareto front differences between and are forcing significant changes in concentrations downstream of the special junctions. These changes are not very large but could lead to a better decision for sensor location when using. MDL-0.1 mg/l MDL-0.01 mg/l Detection Redundancy (%) Detection Redundancy (%) 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 40% 45% 50% 55% 60% 65% 70% 75% 80% MDL mg/l 40% 45% 50% 55% 60% 65% 70% 75% 80% Detection Redundancy (%) Detection Redundancy (%) 100% 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% 0% 40% 45% 50% 55% 60% 65% 70% 75% 80% MDL mg/l 40% 45% 50% 55% 60% 65% 70% 75% 80% Fig. 2. Pareto fronts for Network3, showing the relationship between detection likelihood and detection redundancy for differing MDLs. Note the detection limit and water quality model s effects on the Pareto fronts.

8 458 MDL-0.1 mg/l MDL-0.01 mg/l Detection Time (min) Detection Time (min) % 10% 20% 30% 40% 50% 60% 70% 80% MDL mg/l 0 0% 10% 20% 30% 40% 50% 60% 70% 80% MDL mg/l Detection Time (min) Detection Time (min) % 10% 20% 30% 40% 50% 60% 70% 80% 0 0% 10% 20% 30% 40% 50% 60% 70% 80% Fig. 3. The Pareto fronts for Network 3 showing the relationship between detection likelihood and detection time for alternative MDL. Note the detection limit and water quality model s effects on the Pareto fronts. The Pareto fronts comparing the detection likelihood and detection time objectives are shown in Fig. 3 with alternative MDLs. First, the objectives are in conflict smaller detection time and larger detection likelihoods are desirable. Next, the differences between the two mixing models are not as significant as they were for the previous objectives. Given the size and layout of this network, detection time is not sensitive to small changes, while the plume spread is somewhat more variable. A more highly looped or larger network with more nodes that cause incomplete mixing may produce more significant differences compared to those of this system and should be examined. 4. CONCLUSION Optimal locations for sensor placement in a simple network and under both perfect and imperfect mixing conditions were determined using the multiobjective approach of Preis and Ostfeld (2008). For this network, differences between the two results were not significant. When perfect sensors were assumed (or in any case where concentration is not important but simply the presence of a contaminant), the two mixing models provided very similar results. Differences in the sensor placement designs and the objectives increased with higher (more realistic) MDLs. Although the differences were small in this case, more study should be completed to determine if sensor placement designs would be altered under imperfect mixing conditions in larger and more highly looped networks.

9 REFERENCES Austin, R. G., van Bloemen Waanders, B., McKenna, S., and Choi, C. Y. (2008). Mixing at cross junctions in water distribution systems. II: Experimental study. J. Water Resour. Plann. Manage., 134(3), , Available online at Choi, C.Y., Song, I., and Romero-Gomez, P. (2008a). Solute Transport in a Pressurized Pipe Network with Multiple Cross Junctions. Journal of Hydraulic Engineering, submitted. Choi, C.Y., Shen, J. Y., Austin, R. G. (2008b). Development of a comprehensive solute mixing model () for double-tee, cross, and wye junctions. Proc. 10th Annual Water Distribution System Analysis Symp., Kruger Park, South Africa. Deb K., Pratap A., Agarwal S., and Meyarivan T. (2002). "A fast and elitist multiobjective genetic algorithm: NSGA-II." IEEE Transactions on Evolutionary Computation, Vol. 6, No. 2, pp Lee, B.H. and Deininger R.A. (1992). Optimal locations of monitoring stations in water distribution system. Journal of Environmental Engineering, 118(1), Ostfeld, A., and Salomons, E. (2004). Optimal layout of early warning detection stations for water distribution systems security. J. Water Resour. Plann. Manage., 130(5), Preis, A., and Ostfeld, A. (2008). Multiobjective contaminant sensor network design for water distribution systems. Engineering Optimization, in press. Romero-Gomez, P., Ho, C.K., and Choi, C.Y. (2008a). Mixing at cross junctions in water distribution systems. I: Numerical study. J. Water Resour. Plann. Manage., 134(3), Romero-Gomez, P., Choi, C.Y., Lansey, K., Preis, A., and Ostfeld, A. (2008b). Sensor Network Design with Improved Water Quality Models at Cross Junctions. Proc. 10th Annual Water Distribution System Analysis Symp., Kruger Park, South Africa. van Bloemen Waanders, B., Hammond, G., Shadid, J., Collis, S., and Murray, R. (2005). A comparison of Navier-Stokes and network models to predict chemical transport in municipal water distribution systems. Proc., World Water and Environmental Resources Congress, Anchorage, Alaska. ACKNOWLEDGMENTS This study was supported by the Center for Advancing Microbial Risk Assessment (CAMRA) funded by the U. S. Environmental Protection Agency Science to Achieve Results and the U. S. Department of Homeland Security University Programs grant number R