A Mixed Integer Approach for Obtaining Unique Solutions in Source Inversion of Drinking Water Networks

Abstract

Presidential Decision Directive 63 identified water systems as one of the critical infrastructures to the United States. Following this directive and the passing of The Public Health, Security, and Bioterrorism Preparedness and Response Act there has been increased research effort in both assessing, the vulnerability of drinking water systems and proposing protection measures. Drinking water networks are vulnerable to chemical and biological contamination. While physical security is being used to limit access to some potential contamination locations, due to the distributed nature of drinking water networks, many locations remain unprotected. One proposed method of protection is the installation of an early warning detection system. Sensors installed at various locations throughout the drinking water network could warn utilities companies in the event of a contamination. On its own, an early warning detection system provides only a coarse measure of the time and location of the contamination event. In previous work the authors introduced a large scale nonlinear programming approach that used real-time concentration information from an installed sensor grid to accurately determine the time and location of the contamination event. This approach introduced unknown, time dependent injection terms at every node in the network and formulated a quadratic program to solve for the time profiles of the injections. The objective function was a least squares minimization of the errors between the calculated and measured node concentrations at the sensor nodes with a regularization term to force a unique solution. The constraints in the optimization problem were the partial differential equations of the water quality model for the network. This problem was then discretized with a fully simultaneous approach, using an origin tracking algorithm to characterize the pipe time delays and remove the need to discretize along the length of the pipes. The resulting large scale nonlinear program was solved using a nonlinear interior point code, IPOPT. This approach was effective at identifying a family of possible injection scenarios. The unregularized formulation of the source inversion problem can have many non-unique solutions. The regularized formulation, on the other hand, has a unique solution, but this solution is essentially linear combination of possible injection scenarios. With this approach alone, it is difficult to determine if the observed contamination was caused from a single injection location or multiple locations. In this work, we propose a problem reduction technique and formulate a mixed integer quadratic program (MIQP) to identify unique injection scenarios. This formulation includes constraints that further limit the solution space and allows us to distinguish between single and multiple injection locations. Section 2 gives a brief description of the formulation, followed by a discussion of solution non-uniqueness and how this manifests in the regularized problem. In Section 3 we introduce the mixed integer formulation and show how the problem size can be reduced drastically using active-set information from the original continuous problem. We show the effectiveness of this approach on a real municipal water network in Section 4. Here we test both single and multiple location injection scenarios. Finally, we present some conclusions and directions for future work.