Optimizing Sensor Placements in Water Distribution Systems Using Submodular Function Maximization

Drinking water distribution networks represent complex systems. Water flow rates in a water distribution system vary with time, with periodic features that reflect temporal variations in water demand by consumers. The intentional introduction of a contaminant disrupts the system and could theoretically be detected by a sensor or network of sensors placed at nodes (pipe junctions, reservoirs, storage tanks, or even individual consumer taps) in the system. Determining the best locations for placement of these sensors represents a significant research question, because the system has multiple states, the number of possible intrusion points is large, and the likely high cost of these sensors limits the number that can realistically be deployed. The optimal placement of these sensors to minimize the effect of an introduced contaminant on the population is a critical issue. Sensor placement for intrusion detection exhibits an important diminishing returns property: adding a sensor to a sensor network improves the detection ability less than adding it to a subset of the sensor network. We prove that this submodularity property holds for the objective functions that we consider for placing sensors, and exploit it by applying algorithms for maximizing monotonic submodular functions. Unlike existing optimization algorithms for selecting sensor placements, our efficient optimization procedure has strong theoretical performance guarantees. In spite of the problem’s complexity, our algorithm is guaranteed to always find a solution that is at least within 63% of the optimum, and will often find a (near-)optimal solution. This method is applied to two hypothetical distribution systems (129 nodes and 12,527 nodes) to determine optimal sensor placements for a sensor network of 5 or 20 sensors. Optimization was based on multiple criteria including: (1) minimizing time to detection, (2) minimizing population affected prior to detection, (3) minimizing expected demand for contaminated water prior to detection, and (4) maximizing detection likelihood. A base scenario and three derivative scenarios were used to test the sensor location optimization for the hypothetical systems. In order to compute accurately the objective criteria, we exhaustively simulated all possible attack scenarios, using distributed computation. Five optimization objective functions were considered (i.e., optimization on each of the four objectives independently and then an equally weighted multi-objective optimization). The two networks analyzed in this project illustrate how a sensor network of 20 sensors is more than “adequate” for the example distribution system of 129 nodes, while a much larger sensor network would be needed for “adequate” detection in the example large network of 12,527 nodes. The developed algorithms generalize to networks of arbitrary size and can be constrained by expert knowledge or rankings of scenario likelihood. Further, the optimization algorithms have potential applications for placement of sensors in other complex, dynamic systems.