Robust network sensor location for complete link flow observability under uncertainty

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The link observability problem is to identify the minimum set of links to be installed with sensors that allow the full determination of flows on all the unobserved links. Inevitably, the observed link flows are subject to measurement errors, which will accumulate and propagate in the inference of the unobserved link flows, leading to uncertainty in the inference process. In this paper, we develop a robust network sensor location model for complete link flow observability, while considering the propagation of measurement errors in the link flow inference. Our model development relies on two observations: (1) multiple sensor location schemes exist for the complete inference of the unobserved link flows, and different schemes can have different accumulated variances of the inferred flows as propagated from the measurement errors. (2) Fewer unobserved links involved in the nodal flow conservation equations will have a lower chance of accumulating measurement errors, and hence a lower uncertainty in the inferred link flows. These observations motivate a new way to formulate the sensor location problem. Mathematically, we formulate the problem as min–max and min–sum binary integer linear programs. The objective function minimizes the largest or cumulative number of unobserved links connected to each node, which reduces the chance of incurring higher variances in the inference process. Computationally, the resultant binary integer linear program permits the use of a number of commercial software packages for its globally optimal solution. Furthermore, considering the non-uniqueness of the minimum set of observed links for complete link flow observability, the optimization programs also consider a secondary criterion for selecting the sensor location scheme with the minimum accumulated uncertainty of the complete link flow inference.