William Eugene Hart

FORMULATION AND OPTIMIZATION OF ROBUST SENSOR PLACEMENT PROBLEMS FOR CONTAMINANT WARNING SYSTEMS

The sensor placement problem (SPP) in contaminant warning system (CWS) design for water distribution networks involves maximizing the level of protection afforded by a limited number of sensors. In existing SPP formulations, the protection level is typically quantified as either the expected impact of a contamination event, weighted by occurrence probability, or the proportion of events that are detectable. In these formulations, the issue of how to mitigate against potentially high-impact events is either handled implicitly or ignored entirely. Consequently, any solutions of these formulations run the serious risk of failing to protect against any number of high-impact, 9/11-style attacks. This risk is further amplified by the fact that reliable estimation of contamination event probabilities is extremely difficult, such that existing SPP formulations may significantly discount the potential of high-impact events. In contrast, robust formulations of the SPP directly address these concerns by focusing strictly on a subset of high-impact contamination events, and placing sensors to minimize the impact of such events. We introduce several robust formulations of the SPP that are distinguished by how they quantify the potential damage due to high-impact contamination events. These include minimization of the worst-case impact, the Value at Risk (VaR), and the Tail-Conditional Expectation (TCE). The worst-case formulation is equivalent to the p-center problem in facility location theory. VaR and TCE are standard measures of robustness in the financial literature; the corresponding robust formulations of the SPP respectively minimize the (1-α)% largest impact and a weighted sum of the α% largest impacts. All formulations can be expressed as Mixed-Integer Programs (MIPs), which can be solved using both commercial MIP solvers and specialized heuristics. Additionally, we develop computational methods for exploring the performance trade-offs between robust and expectation-based SPP formulations. We use this framework to explore the nature of robust versus expectation-based solutions to the SPP on three real-world water distribution networks, ranging in size from 400 to over 10,000 junctions. We observe that robust SPP formulations are one or more orders of magnitude more difficult to solve than expectation-based SPPs. Our results indicate that simple heuristics yield optimal solutions to the smaller test problems in shorter run-times than MIP solvers, and yield higher-quality solutions for larger test problems. For realistic sensor budgets, solutions with low expected impact fail to protect against large numbers of high-impact contamination events (with impact 5-10 times larger than the expectation). In contrast, we show that solutions to robust SPPs yield 10-25number and magnitude of high-impact events. In general, our results indicate that it is possible to trade off mean impact versus high impact performance

A FACILITY LOCATION APPROACH TO SENSOR PLACEMENT OPTIMIZATION

1. Overview of Solution Approach The general sensor placement problem (SPP) for contaminant warning system (CWS) design involves placement of a limited number of sensors such that the expected impact of an attack is minimized. We cast the SPP in terms of the well-known p-median problem from discrete location theory. The p-median formulation assumes a fixed number of attack scenarios, each specifying a probability of occurrence, a set of injection sites, injection strengths, and injection durations. The impact of each potential attack is determined via contaminant transport simulation. Specifically, EPANET is used to generate a time-series of contaminant concentration at each network junction for each attack. For each combination of attack and network junction, the resulting time-series are then used to compute the network-wide impact of the attack assuming detection via a hypothetical contaminant sensor placed at the network junction. In conjunction with the attack probabilities, the resulting impact coefficients completely specify the input to our p-median formulation of the SPP. We solve the p-median SPP using domain-specific heuristics based on a combination of GRASP and local search. Extensive computational tests indicate that the heuristics consistently locate globally optimal solutions, as verified via solution of the corresponding mixed-integer program (MIP) using commercially available solvers. Further, the heuristic executes in seconds to minutes for networks ranging from 100 to 10,000 junctions, respectively, subject to large numbers (on the order of the number of network junctions) of hypothetical attack scenarios. By isolating objective-specific information to the impact coefficients, our approach seamlessly allows for optimization of disparate performance objectives, e.g., detection likelihood and population exposed. The use of general-purpose contaminant transport simulators allows us to handle arbitrarily complex attack scenarios, e.g., multiple simultaneous injection sites with different contaminants at variable injection strengths and durations. Certain response strategies, such as delays incurred due to manual verification of sensor hits, can be incorporated via appropriate modification of the impact coefficients. Through the use of side constraints, we are able to generate solutions to the p-median formulation of the SPP that simultaneously yield highquality performance with respect to a range of performance objectives. Finally, in contrast to competing approaches to solving the SPP, we are able to provide provable bounds on the quality of solutions generated by our heuristic via formulation and solution of the SPP as a p-median MIP.

On the Placement of Imperfect Sensors in Municipal Water Networks

We consider the problem of optimally placing water quality sensors in municipal water networks under the assumption that sensors may fail. We give a non-linear formulation of the problem, then a linearization of this formulation in the form of a mixed-integer program (MIP). We explore the scalability limits of this formulation, then use it as a bounding procedure for a local search heuristic that optimizes the same objective: minimizing the expected impact of a contamination event. This heuristic can find optimal or near-optimal solutions on networks with over ten thousand junctions. This paper was presented at the 8th Annual Water Distribution Systems Analysis Symposium which was held with the generous support of Awwa Research Foundation (AwwaRF).

The TEVA-SPOT Toolkit for Drinking Water Contaminant Warning System Design

We present the TEVA-SPOT Toolkit, a sensor placement optimization tool developed within the USEPA TEVA program. The TEVA-SPOT Toolkit provides a sensor placement framework that facilitates research in sensor placement optimization and enables the practical application of sensor placement solvers to real-world CWS design applications. This paper provides an overview of its key features, and then illustrates how this tool can be flexibly applied to solve a variety of different types of sensor placement problems.

Scalable Water Network Sensor Placement via Aggregation

The quality of a sensor placement for a municipal water distribution network is usually judged by its performance across a set of contamination scenarios. As networks grow, the number of scenarios needed to accurately model a full set of possible events based on season, special events, and type of contamination can grow even more rapidly. We introduce two new methods for reducing the problem size. Scenario aggregation introduces no new approximations and can reduce problems size and reduce running time regardless of the solution method. Witness aggregation is a technique well suited for integer-programming-based solution methods. We give two variants of witness aggregation. We present preliminary experimental results for a moderate-sized network and enriched set of scenarios. Applying both scenario and witness aggregation gave a solution within 1% of optimal in two orders of magnitude less time than not using aggregation.

DAKOTA, a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis:version 4.0 reference manual

The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogatebased optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a user’s manual for the Dakota software and provides capability overviews and procedures for software execution, as well as a variety of example studies. Dakota Version 6.11 User’s Manual generated on November 7, 2019

Formulation of Chlorine and Decontamination Booster Station Optimization Problem.

A commonly used indicator of water quality is the amount of residual chlorine in a water distribution system. Chlorine booster stations are often utilized to maintain acceptable levels of residual chlorine throughout the network. In addition, hyper-chlorination has been used to disinfect portions of the distribution system following a pipe break. Consequently, it is natural to use hyper-chlorination via multiple booster stations located throughout a network to mitigate consequences and decontaminate networks after a contamination event. Many researchers have explored different methodologies for optimally locating booster stations in the network for daily operations. In this research, the problem of optimally locating chlorine booster stations to decontaminate following a contamination incident will be described.

Modeling Mathematical Programs with Equilibrium Constraints in Pyomo

We describe new capabilities for modeling MPEC problems within the Pyomo modeling software. These capabilities include new modeling components that represent complementar- ity conditions, modeling transformations for re-expressing models with complementarity con- ditions in other forms, and meta-solvers that apply transformations and numeric optimization solvers to optimize MPEC problems. We illustrate the breadth of Pyomos modeling capabil- ities for MPEC problems, and we describe how Pyomos meta-solvers can perform local and global optimization of MPEC problems.

OPTIMAL DETERMINATION OF GRAB SAMPLE LOCATIONS AND SOURCE INVERSION IN LARGE-SCALE WATER DISTRIBUTION SYSTEMS

We present a mixed-integer linear programming formulation to determine optimal locations for manual grab sampling after the detection of contaminants in a water distribution system. The formulation selects optimal manual grab sample locations that maximize the total pair-wise distinguishability of candidate contamination events. Given an initial contaminant detection location, a source inversion is performed that will eliminate unlikely events resulting in a much smaller set of candidate contamination events. We then propose a cyclical process where optimal grab samples locations are determined and manual grab samples taken. Relying only on YES/NO indicators of the presence of contaminant, source inversion is performed to reduce the set of candidate contamination events. The process is repeated until the number of candidate events is sufficiently small. Case studies testing this process are presented using water network models ranging from 4 to approximately 13000 nodes. The results demonstrate that the contamination event can be identified within a remarkably small number of sampling cycles using very few sampling teams. Furthermore, solution times were reasonable making this formulation suitable for real-time settings.

Water Security Toolkit User Manual Version 1.2.

The Water Security Toolkit (WST) is a suite of open source software tools that can be used by water utilities to create response strategies to reduce the impact of contamination in a water distribution network . WST includes hydraulic and water quality modeling software , optimizati on methodologies , and visualization tools to identify: (1) sensor locations to detect contamination, (2) locations in the network in which the contamination was introduced, (3) hydrants to remove contaminated water from the distribution system, (4) locations in the network to inject decontamination agents to inactivate, remove, or destroy contaminants, (5) locations in the network to take grab sample s to help identify the source of contamination and (6) valves to close in order to isolate contaminate d areas of the network. This user manual describes the different components of WST , along w ith examples and case studies. License Notice The Water Security Toolkit (WST) v.1.2 Copyright c 2012 Sandia Corporation. Under the terms of Contract DE-AC04-94AL85000, there is a non-exclusive license for use of this work by or on behalf of the U.S. government. This software is distributed under the Revised BSD License (see below). In addition, WST leverages a variety ofmorexa0» third-party software packages, which have separate licensing policies: Acro Revised BSD License argparse Python Software Foundation License Boost Boost Software License Coopr Revised BSD License Coverage BSD License Distribute Python Software Foundation License / Zope Public License EPANET Public Domain EPANET-ERD Revised BSD License EPANET-MSX GNU Lesser General Public License (LGPL) v.3 gcovr Revised BSD License GRASP ATT includes randomsample and sideconstraints executable files LZMA SDK Public Domain nose GNU Lesser General Public License (LGPL) v.2.1 ordereddict MIT License pip MIT License PLY BSD License PyEPANET Revised BSD License Pyro MIT License PyUtilib Revised BSD License PyYAML MIT License runpy2 Python Software Foundation License setuptools Python Software Foundation License / Zope Public License six MIT License TinyXML zlib License unittest2 BSD License Utilib Revised BSD License virtualenv MIT License Vol Common Public License vpykit Revised BSD License Additionally, some precompiled WST binary distributions might bundle other third-party executables files: Coliny Revised BSD License (part of Acro project) Dakota GNU Lesser General Public License (LGPL) v.2.1 PICO Revised BSD License (part of Acro project) i Revised BSD License Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. * Neither the name of Sandia National Laboratories nor Sandia Corporation nor the names of its con- tributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS %22AS IS%22 AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IM- PLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL SANDIA CORPORATION BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUD- ING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. ii Acknowledgements This work was supported by the U.S. Environmental Protection Agency through its Office of Research and Development (Interagency Agreement %23 DW8992192801). The material in this document has been subject to technical and policy review by the U.S. EPA, and approved for publication. The views expressed by individual authors, however, are their own, and do not necessarily reflect those of the U.S. Environmental Protection Agency. Mention of trade names, products, or services does not convey official U.S. EPA approval, endorsement, or recommendation. The Water Security Toolkit is an extension of the Threat Ensemble Vulnerability Assessment-Sensor Place- ment Optimization Tool (TEVA-SPOT), which was also developed with funding from the U.S. Environ- mental Protection Agency through its Office of Research and Development (Interagency Agreement %23 DW8992192801). The authors acknowledge the following individuals for their contributions to the devel- opment of TEVA-SPOT: Jonathan Berry (Sandia National Laboratories), Erik Boman (Sandia National Laboratories), Lee Ann Riesen (Sandia National Laboratories), James Uber (University of Cincinnati), and Jean-Paul Watson (Sandia National Laboratories). iii Acronyms ATUS American Time-Use Survey BLAS Basic linear algebra sub-routines CFU Colony-forming unit CVAR Conditional value at risk CWS Contamination warning system EA Evolutionary algorithm EDS Event detection system EPA U.S. Environmental Protection Agency EC Extent of Contamination ERD EPANET results database file GLPK GNU Linear Programming Kit GRASP Greedy randomized adaptive sampling process HEX Hexadecimal HTML HyperText markup language INP EPANET input file LP Linear program MC Mass consumed MILP Mixed integer linear program MIP Mixed integer program MSX Multi-species extension for EPANET NFD Number of failed detections NS Number of sensors NZD Non-zero demand PD Population dosed PE Population exposed PK Population killed TAI Threat assessment input file TCE Tailed-conditioned expectation TD Time to detection TEC Timed extent of contamination TEVA Threat ensemble vulnerability assessment TSB Tryptic soy broth TSG Threat scenario generation file TSI Threat simulation input file VAR Value at risk VC Volume consumed WST Water Security Toolkit YML YAML configuration file format for WST iv Symbols Notation Definition Example %7B , %7D set brackets %7B 1,2,3 %7D means a set containing the values 1,2, and 3. [?] is an element of s [?] S means that s is an element of the set S . [?] for all s = 1 [?] s [?] S means that the statement s = 1 is true for all s in set S . P summation P n i =1 s i means s 1 + s 2 + * * * + s n . %5C set minus S %5C T means the set that contains all those elements of S that are not in set T . %7C given %7C is used to define conditional probability. P ( s %7C t ) means the prob- ability of s occurring given that t occurs. %7C ... %7C cardinality Cardinality of a set is the number of elements of the set. If set S = %7B 2,4,6 %7D , then %7C S %7C = 3. v«xa0less