Non-planar hole-generated networks and link flow observability based on link counters

• Published in: Transportation research. Part B: methodological 2014-10, Vol.68, p.239-261
• Main Authors: Castillo, Enrique, Calviño, Aida, Lo, Hong K., Menéndez, José María, Grande, Zacarías
• Format: Article
• Language: English
• Subjects: Centroids; Equivalence; Flow estimation problem; Joints; Links; Mathematical analysis; Networks; Optimal sensor location; Planar and non-planar networks; Transportation; Vectors (mathematics)
• ISSN: 0191-2615; 1879-2367
• DOI: 10.1016/j.trb.2014.06.015

•We introduce the concept of hole for the first time, with important consequences.•Formulas are given for the upper bound of sensors for full link flow observability.•A method is given to obtain easily subsets of linearly independent path vectors.•A method for a minimum number of counters for full link flow observability is given. The concepts of hole, cycle added link and non-planar hole-generated network are introduced for the first time and used to determine (a) the immediate solution of the node conservation equations in terms of hole and cycle added vectors, and (b) the paths as linear combinations of hole vectors. Two equivalent formulas to obtain the number of links to be observed for complete link observability in non-planar hole-generated networks are given in terms of the numbers of links, nodes, holes, cycle added links and centroid node types. These formulas are applicable without any limitation in the number of centroids and possible link connections. Some simple methods are given to obtain first the maximum number of linearly independent (l.i.) paths and next a minimum set of links to be counted in order to get observability of all link flows. It is demonstrated that the number of l.i. paths in a non-planar hole-generated network coincides with the number of holes and cycle added links in the network and that any path can be obtained by linear combinations of the vectors associated with the hole and cycle added links. The methods are illustrated by their application to several networks.