Publications

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Machine learning models are promising as surrogates in optimization when replacing difficult to solve equations or black-box type models. This work demonstrates the viability of linear model decision trees as piecewise-linear surrogates in decision-making problems. Linear model decision trees can be represented exactly in mixed-integer linear programming (MILP) and mixed-integer quadratic constrained programming (MIQCP) formulations. Furthermore, they can represent discontinuous functions, bringing advantages over neural networks in some cases. We present several formulations using transformations from Generalized Disjunctive Programming (GDP) formulations and modifications of MILP formulations for gradient boosted decision trees (GBDT). We then compare the computational performance of these different MILP and MIQCP representations in an optimization problem and illustrate their use on engineering applications. We observe faster solution times for optimization problems with linear model decision tree surrogates when compared with GBDT surrogates using the Optimization and Machine Learning Toolkit (OMLT).

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This report documents the Resilience Enhancements through Deep Learning Yields (REDLY) project, a three-year effort to improve electrical grid resilience by developing scalable methods for system operators to protect the grid against threats leading to interrupted service or physical damage. The computational complexity and uncertain nature of current real-world contingency analysis presents significant barriers to automated, real-time monitoring. While there has been a significant push to explore the use of accurate, high-performance machine learning (ML) model surrogates to address this gap, their reliability is unclear when deployed in high-consequence applications such as power grid systems. Contemporary optimization techniques used to validate surrogate performance can exploit ML model prediction errors, which necessitates the verification of worst-case performance for the models.

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This manuscript presents the recent advances in Mixed-Integer Nonlinear Programming (MINLP) and Generalized Disjunctive Programming (GDP) with a particular scope for superstructure optimization within Process Systems Engineering (PSE). We present an environment of open-source software packages written in Python and based on the algebraic modeling language Pyomo. These packages include MindtPy, a solver for MINLP that implements decomposition algorithms for such problems, CORAMIN, a toolset for MINLP algorithms providing relaxation generators for nonlinear constraints, Pyomo.GDP, a modeling extension for Generalized Disjunctive Programming that allows users to represent their problem as a GDP natively, and GDPOpt, a collection of algorithms explicitly tailored for GDP problems. Combining these tools has allowed us to solve several problems relevant to PSE, which we have gathered in an easily installable and accessible library, GDPLib. We show two examples of these models and how the flexibility of modeling given by Pyomo.GDP allows for efficient solutions to these complex optimization problems. Finally, we show an example of integrating these tools with the framework IDAES PSE, leading to optimal process synthesis and conceptual design with advanced multi-scale PSE modeling systems.

Neural networks (NN)s have been increasingly proposed as surrogates for approximation of systems with computationally expensive physics for rapid online evaluation or exploration. As these surrogate models are integrated into larger optimization problems used for decision making, there is a need to verify their behavior to ensure adequate performance over the desired parameter space. We extend the ideas of optimization-based neural network verification to provide guarantees of surrogate performance over the feasible optimization space. In doing so, we present formulations to represent neural networks within decision-making problems, and we develop verification approaches that use model constraints to provide increasingly tight error estimates. We demonstrate the capabilities on a simple steady-state reactor design problem.

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This work focuses on estimation of unknown states and parameters in a discrete-time, stochastic, SEIR model using reported case counts and mortality data. An SEIR model is based on classifying individuals with respect to their status in regards to the progression of the disease, where S is the number individuals who remain susceptible to the disease, E is the number of individuals who have been exposed to the disease but not yet infectious, I is the number of individuals who are currently infectious, and R is the number of recovered individuals. For convenience, we include in our notation the number of infections or transmissions, T, that represents the number of individuals transitioning from compartment S to compartment E over a particular interval. Similarly, we use C to represent the number of reported cases.

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We present scalable stochastic optimization approaches for improving power systems' resilience to extreme weather events. We consider both proactive redispatch and transmission line hardening as alternatives for mitigating expected load shed due to extreme weather, resulting in large-scale stochastic linear programs (LPs) and mixed-integer linear programs (MILPs). We solve these stochastic optimization problems with progressive hedging (PH), a parallel, scenario-based decomposition algorithm. Our computational experiments indicate that our proposed method for enhancing power system resilience can provide high-quality solutions efficiently. With up to 128 scenarios on a 2,000-bus network, the operations (redispatch) and investment (hardening) resilience problems can be solved in approximately 6 min and 2 h of wall-clock time, respectively. Additionally, we solve the investment problems with up to 512 scenarios, demonstrating that the approach scales very well with the number of scenarios. Moreover, the method produces high quality solutions that result in statistically significant reductions in expected load shed. Our proposed approach can be augmented to incorporate a variety of other operational and investment resilience strategies, or a combination of such strategies.

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In response to anticipated resource shortfalls related to the treatment and testing of COVID-19, many communities are planning to build additional facilities to increase capacity. These facilities include field hospitals, testing centers, mobile manufacturing units, and distribution centers. In many cases, these facilities are intended to be temporary and are designed to meet an immediate need. When deciding where to place new facilities many factors need to be considered, including the feasibility of potential locations, existing resource availability, anticipated demand, and accessibility between patients and the new facility. In this project, a facility location optimization model was developed to integrate these key pieces of information to help decision makers identify the best place, or places, to build a facility to meet anticipated resource demands. The facility location optimization model uses the location of existing resources and the anticipated resource demand at each location to minimize the distance a patient must travel to get to the resource they need. The optimization formulation is presented below. The model was designed to operate at the county scale, where patients are grouped per county. This assumption can be modified to integrate other scales or include individual patients.

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While peak shaving is commonly used to reduce power costs, chemical process facilities that can reduce power consumption on demand during emergencies (e.g., extreme weather events) bring additional value through improved resilience. For process facilities to effectively negotiate demand response (DR) contracts and make investment decisions regarding flexibility, they need to quantify their additional value to the grid. We present a grid–centric mixed–integer stochastic programming framework to determine the value of DR for improving grid resilience in place of capital investments that can be cost prohibitive for system operators. We formulate problems using both a linear approximation and a nonlinear alternating current power flow model. Our numerical results with both models demonstrate that DR can be used to reduce the capital investment necessary for resilience, increasing the value that chemical process facilities bring through DR. Furthermore, the linearized model often underestimates the amount of DR needed in our case studies.

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In this work, we show that a strong upper bound on the objective of the alternating current optimal power flow (ACOPF) problem can significantly improve the effectiveness of optimization-based bounds tightening (OBBT) on a number of relaxations. We additionally compare the performance of relaxations of the ACOPF problem, including the rectangular form without reference bus constraints, the rectangular form with reference bus constraints, and the polar form. We find that relaxations of the rectangular form significantly strengthen existing relaxations if reference bus constraints are included. Overall, relaxations of the polar form perform the best. However, neither the rectangular nor the polar form dominates the other. In conclusion, with these strategies, we are able to reduce the optimality gap to less than 0.1% on all but 5 NESTA test cases with up to 300 buses by performing OBBT alone.

Here, we show that a strong upper bound on the objective of the alternating current optimal power flow (ACOPF) problem can significantly improve the effectiveness of optimization-based bounds tightening (OBBT) on a number of relaxations. We additionally compare the performance of relaxations of the ACOPF problem, including the rectangular form without reference bus constraints, the rectangular form with reference bus constraints, and the polar form. We find that relaxations of the rectangular form significantly strengthen existing relaxations if reference bus constraints are included. Overall, relaxations of the polar form perform the best. However, neither the rectangular nor the polar form dominates the other. Ultimately, with these strategies, we are able to reduce the optimality gap to less than 0.1% on all but 5 NESTA test cases with up to 300 buses by performing OBBT alone.

We present novel stochastic optimization models to improve power systems resilience to extreme weather events. We consider proactive redispatch, transmission line hardening, and transmission line capacity increases as alternatives for mitigating expected load shed due to extreme weather. Our model is based on linearized or "DC" optimal power flow, similar to models in widespread use by independent system operators (ISOs) and regional transmission operators (RTOs). Our computational experiments indicate that proactive redispatch alone can reduce the expected load shed by as much as 25% relative to standard economic dispatch. This resiliency enhancement strategy requires no capital investments and is implementable by ISOs and RTOs solely through operational adjustments. We additionally demonstrate that transmission line hardening and increases in transmission capacity can, in limited quantities, be effective strategies to further enhance power grid resiliency, although at significant capital investment cost. We perform a cross validation analysis to demonstrate the robustness of proposed recommendations. Our proposed model can be augmented to incorporate a variety of other operational and investment resilience strategies, or combination of such strategies.

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Electricity markets rely on the rapid solution of the optimal power flow (OPF) problem to determine generator power levels and set nodal prices. Traditionally, the OPF problem has been formulated using linearized, approximate models, ignoring nonlinear alternating current (AC) physics. These approaches do not guarantee global optimality or even feasibility in the real ACOPF problem. We introduce an outer-approximation approach to solve the ACOPF problem to global optimality based on alternating solution of upper- and lower-bounding problems. The lower-bounding problem is a piecewise relaxation based on strong second-order cone relaxations of the ACOPF, and these piecewise relaxations are selectively refined at each major iteration through increased variable domain partitioning. Our approach is able to efficiently solve all but one of the test cases considered to an optimality gap below 0.1%. Furthermore, this approach opens the door for global solution of MINLP problems with AC power flow equations.

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The solution of the Optimal Power Flow (OPF) and Unit Commitment (UC) problems (i.e., determining generator schedules and set points that satisfy demands) is critical for efficient and reliable operation of the electricity grid. For computational efficiency, the alternating current OPF (ACOPF) problem is usually formulated with a linearized transmission model, often referred to as the DCOPF problem. However, these linear approximations do not guarantee global optimality or even feasibility for the true nonlinear alternating current (AC) system. Nonlinear AC power flow models can and should be used to improve model fidelity, but successful global solution of problems with these models requires the availability of strong relaxations of the AC optimal power flow constraints. In this paper, we use McCormick envelopes to strengthen the well-known second-order cone (SOC) relaxation of the ACOPF problem. With this improved relaxation, we can further include tight bounds on the voltages at the reference bus, and this paper demonstrates the effectiveness of this for improved bounds tightening. We present results on the optimality gap of both the base SOC relaxation and our Strengthened SOC (SSOC) relaxation for the National Information and Communications Technology Australia (NICTA) Energy System Test Case Archive (NESTA). For the cases where the SOC relaxation yields an optimality gap more than 0.1 %, the SSOC relaxation with bounds tightening further reduces the optimality gap by an average of 67 % and ultimately reduces the optimality gap to less than 0.1 % for 58 % of all the NESTA cases considered. Stronger relaxations enable more efficient global solution of the ACOPF problem and can improve computational efficiency of MINLP problems with AC power flow constraints, e.g., unit commitment.

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Drinking water systems face multiple challenges, including aging infrastructure, water quality concerns, uncertainty in supply and demand, natural disasters, environmental emergencies, and cyber and terrorist attacks. All of these have the potential to disrupt a large portion of a water system causing damage to infrastructure and outages to customers. Increasing resilience to these types of hazards is essential to improving water security. As one of the United States (US) sixteen critical infrastructure sectors, drinking water is a national priority. The National Infrastructure Advisory Council defined infrastructure resilience as “the ability to reduce the magnitude and/or duration of disruptive events. The effectiveness of a resilient infrastructure or enterprise depends upon its ability to anticipate, absorb, adapt to, and/or rapidly recover from a potentially disruptive event”. Being able to predict how drinking water systems will perform during disruptive incidents and understanding how to best absorb, recover from, and more successfully adapt to such incidents can help enhance resilience.

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Water utilities are vulnerable to a wide variety of human-caused and natural disasters. The Water Network Tool for Resilience (WNTR) is a new open source Python™ package designed to help water utilities investigate resilience of water distribution systems to hazards and evaluate resilience-enhancing actions. In this paper, the WNTR modeling framework is presented and a case study is described that uses WNTR to simulate the effects of an earthquake on a water distribution system. The case study illustrates that the severity of damage is not only a function of system integrity and earthquake magnitude, but also of the available resources and repair strategies used to return the system to normal operating conditions. While earthquakes are particularly concerning since buried water distribution pipelines are highly susceptible to damage, the software framework can be applied to other types of hazards, including power outages and contamination incidents.