Elsevier

Automatica

Volume 49, Issue 11, November 2013, Pages 3351-3358
Automatica

Brief paper
Optimal coverage of an infrastructure network using sensors with distance-decaying sensing quality

https://doi.org/10.1016/j.automatica.2013.07.029Get rights and content

Abstract

Motivated by recent applications of wireless sensor networks in monitoring infrastructure networks, we address the problem of optimal coverage of infrastructure networks using sensors whose sensing performance decays with distance. We show that this problem can be formulated as a continuous p-median problem on networks. The literature has addressed the discrete p-median problem on networks and in continuum domains, and the continuous p-median problem in continuum domains extensively. However, in-depth analysis of the continuous p-median problem on networks has been lacking. With the sensing performance model that decays with distance, each sensor covers a region equivalent to its Voronoi partition on the network in terms of the shortest path distance metric. Using Voronoi partitions, we define a directional partial derivative of the coverage metric with respect to a sensor’s location. We then propose a gradient descent algorithm to obtain a locally optimal solution with guaranteed convergence. The quality of an optimal solution depends on the choice of the initial configuration of sensors. We obtain an initial configuration using two approaches: by solving the discrete p-median problem on a lumped network and by random sampling. We consider two methods of random sampling: uniform sampling and D2-sampling. The first approach with the initial solution of the discrete p-median problem leads to the best coverage performance for large networks, but at the cost of high running time. We also observe that the gradient descent on the initial solution with the D2-sampling method yields a solution that is within at most 7% of the previous solution and with much shorter running time.

Introduction

With recent advances in wireless sensor networking technologies, it is now possible to monitor built infrastructures at scales not possible five years ago. There is a growing interest in instrumenting critical urban infrastructure networks such as road networks, water or gas pipe networks and electric grids for monitoring and improving their operations and security. A fundamental question in instrumenting an infrastructure is sensor coverage, i.e., where to place the sensors and how often to collect data. Coverage captures the notion of overall quality-of-sensing. In this paper we address the problem of optimal coverage of an infrastructure network with sensors for which the sensing performance decays monotonically with the distance from them. Our problem is inspired by recent work on detecting bursts and leaks in a water distribution network using pressure sensors (Misiunas, 2005, Srirangarajan, 2010, Whittle, 2010). A sudden pipe burst creates pressure transients which travel along the network and are seen as drops in pressure values by pressure sensors (Misiunas, 2005). The magnitude of a pressure transient decays with distance due to friction and partial reflection at pipe junctions. As a result, beyond some distance, the pressure drop may appear insignificant. Thus, the overall performance of burst detection depends on the number of pressure sensors and their placement in the network, i.e., the sensor coverage. Another example where sensor coverage is important is in preparing against a possible security threat to the water supply system through a deliberate injection of chemical or biological contaminant (Krause, 2008, Ostfield, 2008, Phillips et al., 2006). The goal here is to optimize coverage of the water distribution network using chemical sensors to minimize the expected time of contaminant detection and the impact on the population (Ostfield, 2008).

We formulate the sensor coverage problem on networks as a locational optimization problem or a facility location problem. Operations research literature has extensively addressed the facility location problem and its variations in continuum as well as in network domains (Francis et al., 1983, Hakimi, 1964, Okabe and Suzuki, 1997, Owen and Daskin, 1998, Plastria, 2001). The facility location problem involves finding optimal locations of facilities given the demand distribution so as to minimize the worst-case or average distance to the nearest facility. The variations discussed in the literature depend on the nature of demand distributions (discrete or continuous), candidate facility locations (discrete or continuous), type of domain (continuum or network), shape of facilities (point or area) and objectives of facilities (distance minimizing and/or competing) (Francis et al., 1983, Hakimi, 1964, Klose and Drexl, 2005, Okabe and Suzuki, 1997, Owen and Daskin, 1998, Plastria, 2001, ReVelle and Eiselt, 2005).

In network domains, researchers have mostly considered discrete demand distributions and discrete candidate facility locations (Francis et al., 1983, Klose and Drexl, 2005, Okabe and Suzuki, 1997, ReVelle and Eiselt, 2005). This variation of the problem maps into the well-known discrete p-center or discrete p-median problems depending on whether we want to minimize the worst-case or average distance to the nearest facilities. Both the problems have been shown to be NP-hard and can be formulated as mixed integer linear programs (MILPs) (e.g. see survey  ReVelle & Eiselt, 2005). We formulate our sensor coverage problem on the network as a continuous p-median problem involving continuous demand distributions and continuous candidate sensor locations anywhere on the network. Okabe and Boots note in their survey (Okabe & Suzuki, 1997) that the continuous p-median problem is seldom addressed in network domains. This problem is usually considered in continuum domains. The optimal solution in continuum domains is obtained by the centroidal Voronoi partitioning, where each facility is at the centroid of its Voronoi partition or coverage region (Okabe & Suzuki, 1997). One approach to find an optimal solution is to use Lloyd’s gradient descent algorithm starting with the initial placement of sensors (Okabe & Suzuki, 1997). In this iterative procedure, a facility is incrementally moved toward the centroid of its current Voronoi partition with step size proportional to the local gradient of the overall coverage metric. This algorithm is guaranteed to converge to a locally optimal solution (Okabe & Suzuki, 1997). Since the computation of the Voronoi partition depends only on the locations of the neighboring facilities, this approach has also found applications in coverage with distributed mobile robotic sensor networks with decentralized control (Cortes et al., 2004, Deshpande et al., 2009, Lekien and Leonard, 2009, Schwager et al., 2008, Schwager et al., 2007).

We show with the help of examples that in many cases the continuous p-median problem formulation has advantages over the discrete p-median formulation as the overall coverage metric is improved. In principle, we can finely discretize the network, lump continuous demand and candidate sensor locations into discrete points, and formulate and solve the discrete p-median problem using MILPs. However, for large networks, we show that this greatly increases the complexity of the problem and running time. Instead, we propose an efficient gradient descent algorithm to solve the problem. We extend the notion of Voronoi partitioning on networks in terms of the shortest path distances. Using this, we define the notion of a gradient of the objective function or the coverage metric with respect to a sensor location. The main challenge lies in defining gradients at the network junctions or nodes, where the coverage metric can be non-smooth. We define gradients at a junction in a special manner using perturbed locations along the emanating edges. We then propose a gradient descent algorithm to obtain a locally optimal solution. This iterative approach requires an initial solution. We obtain it using two approaches. In the first approach, we obtain an initial sensor placement by solving the discrete p-median problem with coarse discretization of the network. In the second approach, we use random sampling to find an initial sensor placement. Here we use two methods: D2-sampling and uniform sampling. While the first approach yields the best coverage performance for large networks in our examples, we also observe that the best case D2-sampling over 25 runs yields coverage metric that is not more than 7% of the prior solution and, that too, with much smaller running time.

Section snippets

Problem formulation

We represent the infrastructure network as an undirected graph G(V,E), where V and E correspond to the set of vertices and edges respectively. An infrastructure network is directional (e.g., water flow and electricity flow in networks). However, the change associated with the event we are trying to sense may or may not be directional. For instance, a pressure transient wave due to a sudden pipe burst travels in all directions, whereas, water contaminants only travel downstream. We focus on the

Gradient descent algorithm

In this section we describe our gradient descent algorithm to find an optimal solution to the continuous p-median problem. The algorithm starts with an initial configuration of the sensors. The main idea of the algorithm is that in every iteration, each sensor moves to a new position by the distance proportional to its local gradient of the objective function H. When we update each sensor location, we consider two cases depending on whether the sensor is in the middle of an edge or is at a

Initial sensor placement

The gradient descent algorithm described in the previous section requires an initial configuration of sensors. Depending on the initial configuration, the algorithm may lead to a better or worse local optimal solution. We describe two approaches to obtain an initial sensor configuration. In the first approach we cast the continuous p-median problem into the discrete p-median problem, solve it using an MILP formulation and use its solution as an initial configuration. In the second approach, we

Simulation results

For all of our simulations, we focus on the case where the sensing cost function is the shortest path distance, i.e., ϕ(q)=1 and f(dpq)=dpq. We implemented our gradient descent algorithm in the MATLAB environment. We use MATLAB’s in-built shortest path functionality to find the shortest paths between all pairs of vertices once in the beginning of the simulation. This is useful in updating Voronoi partitions and Voronoi neighbors of sensors as their locations change. In all the simulations, we

Conclusions and future work

We have addressed the optimal sensor coverage of infrastructure networks for sensors whose sensing performance decays with the distance. We formulated the problem as a continuous p-median problem on networks. Using Voronoi partitions, we defined a directional gradient of the coverage metric with respect to a sensor location. We then proposed a gradient descent algorithm to obtain a locally optimal solution with guaranteed convergence. We discussed simulation results for various network

Acknowledgments

The first author would also like to thank Dr. Prahladh Harsha for pointing to the recent work in D2-sampling, and Dr. Pavithra Harsha for reviewing the MILP formulation implemented in this work.

Ajay Deshpande is a Research Staff Member at the IBM T.J. Watson Research Center in Yorktown Heights, NY. Previously, he also worked as a Postdoctoral Associate in the Laboratory for Manufacturing and Productivity at the Massachusetts Institute of Technology. He received his Ph.D in Mechanical Engineering from MIT in 2008. He also earned his double Masters in Mechanical Engineering, and Electrical Engineering and Computer Science from MIT in 2006. He was a recipient of the MIT Presidential

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Ajay Deshpande is a Research Staff Member at the IBM T.J. Watson Research Center in Yorktown Heights, NY. Previously, he also worked as a Postdoctoral Associate in the Laboratory for Manufacturing and Productivity at the Massachusetts Institute of Technology. He received his Ph.D in Mechanical Engineering from MIT in 2008. He also earned his double Masters in Mechanical Engineering, and Electrical Engineering and Computer Science from MIT in 2006. He was a recipient of the MIT Presidential Fellowship. He graduated from the Indian Institute of Technology Bombay in 2001 with his B.Tech and M.Tech.

Sanjay Sarma is the Fred Fort Flowers (1941) and Daniel Fort Flowers (1941) Professor of Mechanical Engineering at MIT. He co-founded the Auto-ID Center at MIT and developed many of the key technologies behind the EPC suite of RFID standards now used worldwide. Dr. Sarma received his Bachelors from the Indian Institute of Technology, his Masters from Carnegie Mellon University and his Ph.D. from the University of California at Berkeley. Sarma also worked at Schlumberger Oilfield Services in Aberdeen, UK, and at the Lawrence Berkeley Laboratories in Berkeley, California.

Kamal Youcef-Toumi joined the MIT Mechanical Engineering Department faculty in 1986. He earned his advanced degrees (M.S. 1981 and Sc.D. 1985) in Mechanical Engineering from MIT. His undergraduate degree (B.S. in Mechanical Engineering awarded in 1979) is from the University of Cincinnati.

Professor Youcef-Toumi’s research has focused primarily on design, modeling, simulation, instrumentation, and control theory. The applications have included manufacturing, robotics, automation, metrology and nano/biotechnology. He teaches courses in the fields of dynamic systems and controls, robotics, and precision machine design and automatic control systems. He is the Co-Director of the Center for Clean Water and Clean Energy at MIT and KFUPM.

Professor Youcef-Toumi was selected as a National Science Foundation Presidential Young Investigator “in recognition of research and teaching accomplishments and academic potential”. He has served as a consultant for a many companies including AT&T Bell Laboratories, Mitsubishi ElectricCorp, Penn State University-College of Medicine—Cancer Institute.

Professor Youcef-Toumi is the author of over 175 publications, including a textbook on the theory and practice of direct-drive robots. He holds many patents. Professor Youcef-Toumi has been an invited lecturer at over 135 seminars at companies, research centers and universities throughout the world. He is a member of IEEE and is an ASME Fellow.

Samir Mekid is a Chartered Engineer and Associate Professor graduating from National Polytechnic School (Algiers) in 1988, and then received his M.Sc. and Ph.D. from Compiegne University in France. He has published over 140+ publications in professional journals and international conference proceedings and recently edited two books. He holds several patents. He joined UMIST (UK) in 2001 as Assistant Professor after he worked for Caterpillar in France. He moved in 2008 to King Fahd University of Petroleum & Minerals (KFUPM) in Dhahran, Saudi Arabia. His research areas are in Precision Machine Design, Instrumentation, Measurements and Robotics. He has been invited as European evaluator for several EU research proposals and reviewer to many journals.

The authors would like to thank the King Fahd University of Petroleum and Minerals in Dhahran, Saudi Arabia, for funding the research reported in this paper through the Center for Clean Water and Clean Energy at MIT and KFUPM. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Giancarlo Ferrari-Trecate under the direction of Editor Ian R. Petersen.

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