Production, Manufacturing and Logistics
Robust placement of sensors in dynamic water distribution systems

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Abstract

Designing a robust sensor network to detect accidental contaminants in water distribution systems is a challenge given the uncertain nature of the contamination events (what, how much, when, where and for how long) and the dynamic nature of water distribution systems (driven by the random consumption of consumers). We formulate a set of scenario-based minimax and minimax regret models in order to provide robust sensor-placement schemes that perform well under all realizable contamination scenarios, and thus protect water consumers. Single-and multi-objective versions of these models are then applied to a real water distribution system. A heuristic solution method is applied to solve the robust models. The concept of “sensitivity region” is used to visualize trade-offs between multiple objectives.

Introduction

Safe drinking water is vital for human health and well-being. Accidental water contamination could pose great risk for the consumers. An outbreak of acute watery diarrhea in Milwaukee, US, in 1993 affected more than 400,000 people when a microorganism was transported in the distribution system (Mac Kenzie et al., 1994). An outbreak of waterborne disease epidemic in Walkerton, Ontario, Canada, in 2000 affected 2,300 people as a result of exposure to contaminated drinking water (Hrudey et al., 2003). However, routine periodic monitoring of water quality as required by US Environmental Protection Agency (EPA) does not meet the need of early warning for these types of contamination events. The design of a monitoring system that can detect contaminants in real time is challenging for technical and operational reasons. Advances in chemical and biological sensor technology (Diamond, 1998, Hashemian, 2005) have made it feasible to build such a monitoring system. Ideally, we would place sensors at each node of the distribution network to give the earliest possible warning to the full population. However, sensors are generally expensive (e.g., a Hach chlorine sensor costs between USD 3,000–5,000), so the “optimal” placement of an affordable number of sensors needs to be considered.

The goal of sensor-placement in water distribution systems is to develop a proactive approach to detect contamination events and thus to mitigate the impact of contamination events. A key premise is that a contamination event might occur. However, there is uncertainty as to what the contaminant could be (States et al., 2003), as well as the locations, timing, and duration of the event. Furthermore, even if we knew all these details, there is still uncertainty about the contaminant behavior in the system because of the dynamic nature of the water flows, as represented by the temporal variation of flow rates and directions. The dynamic nature of water flows is driven by pumping rates and pressure applied by the water utility at different entry points to the system as well as consumption patterns of consumers, which exhibits daily, weekly and seasonal variation. The combinations of uncertainties inherent in details of a contamination event and the dynamic nature of water distribution systems pose a challenge in providing sensor-placement schemes that work well for all realizable contamination scenarios.

Placing sensors in water distribution systems is a facility location problem, in which sensors act as facilities and water distribution systems are the environments. Significant research has taken place to develop optimization models for making facility location decisions subject to financial, physical and policy constraints. For a thorough review on this research fields, readers are referred to Chung, 1986, Brandeau and Chiu, 1989, Owen and Daskin, 1998, Current et al., 1990, ReVelle and Eiselt, 2005, Snyder, 2006. Here, we summarize the facility location models that have been deployed to address the problem of locating sensors in water distribution systems.

Three types of decision models are commonly used to design strategies for placing sensors in water distribution systems. The first is deterministic optimization that is based on either a single hypothetical instance assumed to be representative of the behavior of a water distribution system (Kessler et al., 1998; Kumar et al., 1999), or the expected value of the input data across all contamination scenarios (Krause et al., 2006). The second is stochastic optimization that optimizes the expected value of an objective (e.g., minimizing the expected volume of contaminated water consumed prior to detection) (Berry et al., 2006). The third is robust optimization that minimizes either the worst-case scenario or a set of high-impact events depending on how robust is defined (Carr et al., 2006, Watson et al., 2006). Table 1 summarizes the recent research literature which proposes these decision models to place sensors in water distribution systems (for each category, a few sample papers are listed).

In previous research, neither deterministic models nor stochastic models meet the goal of minimizing the losses in health and life of the population across all realizable contamination scenarios. The optimal solution to a deterministic model performs well for a specific instance, but it could be significantly sub-optimal if other scenarios are realized. A stochastic optimization model accounts for the many contamination scenarios, but requires the specification of the probability of each of the scenarios. It is questionable whether meaningful probabilities can be assigned to random variables about which we have inadequate knowledge. Even if the probability of each of the scenarios can be estimated, the solution is optimal for the expected value of the objective function and more often cannot be used to hedge against the worst scenarios. The robust model formulated by Watson et al., 2006, Carr et al., 2006 focuses on addressing uncertainties related to the contamination location (location uncertainty) and does not explicitly incorporate the uncertainties related to the contamination occurrence time (temporal uncertainty). Given the fact that water flow is driven by the population’s consumption and the population consumption pattern shows temporal variations, different contamination occurrence times imply different consequent impacts. For example, the impact of a mid-night contamination event is expected to be different from the impact of a contamination event occurred at noon. In this paper, we formulate two sets of robust models for designing sensor-placement schemes in water distribution systems to address temporal uncertainty. The first set of models is intended to protect as large a population as possible. The second set of models identifies as many contamination events as possible (maximize detection likelihood). We then compare the performance of these models with deterministic models that use the expected value of the input data across all the contamination scenarios. Since robust models tend to be computationally costly, we need to show a clear performance advantage to justify their use. Also, none of the above-mentioned robust optimization models address the multi-objective nature of this sensor-placement problem. We then propose a multi-objective model to maximize detection likelihood and minimize the population at risk as Krause et al. (2006) shows that these two objectives conflict.

Robust optimization addresses uncertainties in problem-input parameters through scenario planning and minimizes losses associated with the worst scenario. It has been applied in diverse domains, including logistic management (Yu and Li, 2000), scheduling (Lebedev and Averbakh, 2006), and facility location (Serra and Marianov, 1998, Burkard and Dollani, 2001). The prominent feature of the robust optimization approach is that a chosen solution performs well even in the worst case. Common robustness criteria include absolute robustness, robust deviation, and relative robustness (Kouvelis and Yu, 1997). The approach based on absolute robustness is also called the minimax approach, while the methods based on robust deviation and relative robustness are called minimax regret approaches. The difference between the latter two lies in the way regret is defined. With a robust deviation criterion, the regret for a scenario is defined as the difference between (1) the performance of the solution for the robust minimax regret approach if scenario s is realized and (2) the performance of the optimal solution to the deterministic model using scenario s as the instance feed. With a relative robust criterion, the regret as defined above is modified by dividing it by the performance of the robust minimax solution. In the minimax (absolute robustness) approach, feasible solutions are evaluated across all scenarios, and the solution that performs the best in the worst scenario is the robust one. For the minimax regret approach, feasible solutions are evaluated across all scenarios, and the solution that performs the best based on the regret criterion in the worst scenario is the robust solution. The choice of the criteria should be driven by the specific need (Kouvelis and Yu, 1997). In general, the minimax approach tends to be very conservative and tries to hedge against the worst scenario while minimax regret approach is less conservative and tries to seek opportunities for improvement in decision making. In this paper, we use both criteria.

Solving robust optimization models are more difficult than the corresponding deterministic models. For a detailed review on solving robust optimization models, see Snyder (2005) and Kouvelis and Yu (1997). Generally, optimal results for robust facility location models are obtained for problems with special structure, such as the 1-median problem (Chen and Lin, 1998) and the 1-center problem on trees (Burkard and Dollani, 2002); for more general problems, researchers tend to resort to heuristic algorithms. In our research, a heuristic solution method is applied to solve the robust models, which is computationally affordable to most (if not all) water utilities. The concept of “sensitivity region” (Gunawan and Azarm, 2005) is used to generate the trade-off relationship in multiple-objective optimization.

In the next section, we provide a detailed description of the problem formulation. Section 3 then gives a brief introduction to the solution methods to this type of robust models. Section 4 applies the robust models to place sensors in a real-world water distribution system. Finally, we conclude with a discussion of future research directions.

Section snippets

Problem formulation

A sensor network in a water distribution system is expected to (1) protect as large a population as possible, (2) identify as many contamination events as possible, and (3) detect contamination events as quickly as possible to allow for a timely response. These objectives have been used by many authors, either singly or in combination (Berry et al., 2005, Kumar et al., 1999, Ostfeld and Salomons, 2005). In addition, the volume of contaminated water consumed prior to detection is also used as an

Solution method

We start by solving the single-objective models, and then solve the multi-objective model. Since the deterministic maximum covering problem and p-median problem are known to be NP-hard for a general network (Drezner and Hamacher, 2002), the corresponding robust optimization models are necessarily NP-hard as well (Snyder, 2006). Serra and Marianov (1998) formulated a robust p-median-based model for locating fire stations in Barcelona to minimize the maximum population-weighted travel time across

Application to a water distribution system

Applications of our robust optimization models to place sensors in a water distribution system require knowledge of how water travels and contaminant behaves in the system. Since it is not possible to conduct contamination experiments in a real distribution system, researchers often model the water distribution system and the associated water behavior and then work with simulation results rather than using experimental data. With such models (hydraulic and water quality models), we are able to

Conclusion and next steps

When placing sensors to detect contaminants in water distribution systems, commonly used decision models (deterministic optimization models and stochastic optimization models) fail to meet the goal of minimizing the life and health of the population across all realizable contamination scenarios. Robust optimization as an approach to hedge against the worst consequences provides a promising basis for locating sensors in water distribution systems to detect contamination events. Current robust

Acknowledgement

This work was funded by the National Science Foundation under grant BES-0329549. We wish to thank Andreas Krause for providing the simulation results on which our analysis was based, as well as James Uber and Avi Ostfeld for providing the network.

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