Synergistic sensor location for link flow inference without path enumeration: A node-based approach

Abstract

Sensors are becoming increasingly critical elements in contemporary transportation systems, gathering essential (real-time) traffic information for the planning, management and control of these complex systems. In a recent paper, Hu, Peeta and Chu introduced the interesting problem of determining the smallest subset of links in a traffic network for counting sensor installation, in such a way that it becomes possible to infer the flows on all remaining links. The problem is particularly elegant because of its limited number of assumptions. Unfortunately, path enumeration was required, which – as recognized by the authors – is infeasible for large-scale networks without further simplifying assumptions (that would destroy the assumption-free nature of the problem). In this paper, we present a reformulation of this link observability problem, requiring only node enumeration. Using this node-based approach, we prove a conjecture made by Hu, Peeta and Chu by deriving an explicit relationship between the number of nodes and links in a transportation network, and the minimum number of sensors to install in order to be able to infer all link flows. In addition, we demonstrate how the proposed method can be employed for road networks that already have sensors installed on them. Numerical examples are presented throughout.

Introduction

With the advent of Intelligent Transportation Systems (ITS), sensors are becoming increasingly critical elements of modern transportation systems (e.g. see Ban et al., 2009, Ng and Waller, 2010a). This growing need for (real-time) traffic information has resulted in an interesting class of problems collectively known as sensor location problems. In these problems, sensors are optimally located for purposes such as origin–destination (O–D) matrix estimation (e.g. see Yang and Zhou, 1998, Bianco et al., 2001, Chootinan et al., 2005, Gan et al., 2005, Castillo et al., 2008a, Mínguez et al., 2010), path flow estimation (e.g. see Gentili and Mirchandani, 2005, Castillo et al., 2008b) and, more recently, link flow inference (e.g., Hu et al., 2009, Castillo et al., 2010).

In a recent paper, Hu et al. (2009) introduced the rather interesting problem of determining the smallest subset of links in a network for counting sensors installation, in such a way that it becomes possible to infer the flows on all remaining, non-equipped links (this problem is referred to by Castillo et al., 2010 as the link observability problem). As the authors pointed out, numerous applications can potentially benefit from the ability to determine all link flows in a network, including dynamic traffic assignment (e.g. see Peeta and Ziliaskopoulos, 2001), evacuation planning (e.g. see Ng and Waller, 2010b) and pavement management systems (e.g. see Ng et al., 2011a). In addition, link flow measurements can also be tremendously useful in travel time reliability studies (e.g. see Chen et al., 2002, Lo et al., 2006, Ng and Waller, 2010c, Ng et al., 2011b). The proposed problem was particularly elegant because of its assumption-free character. Particularly, the link observability problem does not require any assumptions on turning proportions, rout choice behavior and O–D matrices, information that is hard – if not impossible – to obtain in practice. Unfortunately, the problem maintained the restrictive requirement of path enumeration that is common in traffic observability problems (Castillo et al., 2008c). In large-scale, real-world networks, it is clearly impossible to enumerate all paths. In such cases, one needs to resort to simplifying assumptions such as “most likely paths” (as indicated by Hu et al. (2009)), destroying the assumption-free nature of the problem. Path enumeration is particularly vexing since at this time it is not clear at all how sensor location decisions for large-scale networks are affected by the failure to enumerate every single path in a network.

The main contributions of this paper are as follows. First, we demonstrate that the link observability problem can be reformulated using a node-based approach. That is, instead of path enumeration, we only require node enumeration, which is obviously a much simpler task for large-scale networks. Second, using this alternative modeling approach, we prove (and refine) the conjecture made in Hu et al. (2009) that “there may be an upper bound on the number of basis links that is governed by the network topology irrespective of the total number of links in the network”. More specifically, we explicitly state this upper bound (see Proposition 2) and show that it is dependent on the total number of links. Third, recognizing that road networks typically already have sensors installed on them, it is demonstrated how the set of inferable link flows can be determined using a node-based approach (Castillo et al., 2010 addressed this question in terms of the path-link incidence matrix), and how all link flows can be made observable though the addition of the smallest number of sensors.

The remainder of this paper is organized as follows. In Section 2, the main results of the paper are developed. In addition to the numerical examples in Section 2, Section 3 provides some more case studies to demonstrate the proposed methods. Section 4 concludes and summarizes the main results in this paper.