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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Decomposing Loosely Coupled Mixed-Integer Programs for Optimal Microgrid Design

Microgrids are frequently employed in remote regions, in part because access to a larger electric grid is impossible, difficult, or compromises reliability and independence. Although small microgrids often employ spot generation, in which a diesel generator is attached directly to a load, microgrids that combine these individual loads and augment generators with photovoltaic cells and batteries as a distributed energy system are emerging as a safer, less costly alternative. In this work, we present a model that seeks the minimum-cost microgrid design and ideal dispatched power to support a small remote site for one year with hourly fidelity under a detailed battery model; this mixed-integer nonlinear program (MINLP) is intractable with commercial solvers but loosely coupled with respect to time. A mixed-integer linear program (MIP) approximates the model, and a partitioning scheme linearizes the bilinear terms. We introduce a novel policy for loosely coupled MIPs in which the system reverts to equivalent conditions at regular time intervals; this separates the problem into subproblems that we solve in parallel. We obtain solutions within 5% of optimality in at most six minutes across 14 MIP instances from the literature and solutions within 5% of optimality to the MINLP instances within 20 minutes.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Parallel Solver Framework for Mixed-Integer PDE-Constrained Optimization

ROL-PEBBL is a C++, MPI-based parallel code for mixed-integer PDE-constrained optimization (MIPDECO). In these problems we wish to optimize (control, design, etc.) physical systems, which must obey the laws of physics, when some of the decision variables must take integer values. ROL-PEBBL combines a code to efficiently search over integer choices (PEBBL = Parallel Enumeration Branch-and-Bound Library) and a code for efficient nonlinear optimization, including PDE-constrained optimization (ROL = Rapid Optimization Library). In this report, we summarize the design of ROL-PEBBL and initial applications/results. For an artificial source-inversion problem, finding sources of pollution on a grid from sparse samples, ROL-PEBBLs solution for the nest grid gave the best optimization guarantee for any general solver that gives both a solution and a quality guarantee.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

McCormick envelopes in mixed-integer PDE-constrained optimization

McCormick envelopes are a standard tool for deriving convex relaxations of optimization problems that involve polynomial terms. Such McCormick relaxations provide lower bounds, for example, in branch-and-bound procedures for mixed-integer nonlinear programs but have not gained much attention in PDE-constrained optimization so far. This lack of attention may be due to the distributed nature of such problems, which on the one hand leads to infinitely many linear constraints (generally state constraints that may be difficult to handle) in addition to the state equation for a pointwise formulation of the McCormick envelopes and renders bound-tightening procedures that successively improve the resulting convex relaxations computationally intractable. We analyze McCormick envelopes for a model problem class that is governed by a semilinear PDE involving a bilinearity and integrality constraints. We approximate the nonlinearity and in turn the McCormick envelopes by averaging the involved terms over the cells of a partition of the computational domain on which the PDE is defined. This yields convex relaxations that underestimate the original problem up to an a priori error estimate that depends on the mesh size of the discretization. These approximate McCormick relaxations can be improved by means of an optimization-based bound-tightening procedure. We show that their minimizers converge to minimizers to a limit problem with a pointwise formulation of the McCormick envelopes when driving the mesh size to zero. We provide a computational example, for which we certify all of our imposed assumptions. The results point to both the potential of the methodology and the gaps in the research that need to be closed. Our methodology provides a framework first for obtaining pointwise underestimators for nonconvexities and second for approximating them with finitely many linear inequalities in an infinite-dimensional setting.

Approximations and Expansions↗
DOE OSTI2025

Tightest Mixed-Integer Programming Formulations for Quadratic SCUC Optimization

In this project, we developed new, tighter Mixed-Integer Programming (MIP) formulations for the combined Alternating Current (AC) Security-Constrained Unit Commitment (SCUC) and Security-Constrained Optimal Power Flow (SCOPF). The work addresses a critical challenge in power system operations: efficiently determining which generation units to commit and how to optimally dispatch them while maintaining network reliability constraints for both normal and contingency scenarios. Our efforts: 1. Advance the Understanding of SCUC/SCOPF Modeling: By introducing tighter MIP formulations and leveraging cutting-edge optimization tools (Julia/JuMP, PowerModels.jl), this project has pushed forward the state of the art in efficient power systems scheduling. 2. Enhance Technical and Economic Feasibility: The methods developed provide more accurate and potentially faster solutions to large-scale, realistic scheduling and dispatch problems in electric power systems, which can translate into improved reliability and potentially lower costs for grid operations. 3. Benefit to the Public: Greater efficiency in power system operations leads to cost savings for utilities and end-users. Improved reliability and integration of advanced modeling approaches can facilitate the adoption of clean energy resources and better accommodate uncertainties in renewable generation. Because this technology could impact bulk power markets and reliability, these innovations have far-reaching public benefits in terms of cost savings, reliability, and sustainability.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2021

Stochastic Continuous-time Flexibility Scheduling and Pricing in Wholesale Electricity Markets

Large-scale integration of intermittent renewable energy sources (RES) is calling for additional flexibility resources as well as more advanced modeling and optimization techniques to account for the increasing uncertainty and variability in power systems operation. As the RES integration gains momentum, the magnitude and frequency of their variations increase, which may trigger ramping scarcity events in real-time power systems operation. This necessitates revisiting the present definition of power systems flexibility and reserve services to reflect their robustness and adequacy towards sub-interval variations of the load and RES, as well as adjusting the operation models to accommodate the new reserve services. This project took a fundamental approach and aimed at developing continuous-time scheduling and pricing model that accurately models the continuous-time variations of load and RES and efficiently deploys the ramping capability of flexible resources to compensate the sources of variability and uncertainty in the market. In this regard, this project pursued the following goals: Developing stochastic multi-fidelity continuous-time optimization models for scheduling of energy storage (ES) systems and flexible loads in wholesale energy markets; Developing the theory and practices of continuous-time locational marginal pricing for valuating energy storage systems and flexible loads in wholesale energy markets; Developing function space solution approach to convert the proposed stochastic multi-fidelity continuous-time optimization models into tractable mixed-integer linear optimization models; and Defining flexibility reserve as a new type of reserve in markets that would enable ultimate participation of energy storage devices in provision of services to compensate the variability and uncertainty of RES in electricity markets. This project successfully completed all five major tasks defined in the SOPO, and produced 8 high-impact journal papers, 6 conference papers, 3 published U.S. patents, and one web-based software for continuous-time operation optimization of power systems. The application of the proposed flexibility reserve and the stochastic multi-fidelity continuous-time operation scheduling models would modify the forward commitment and schedule of generating units, ES devices and flexible loads, and would line up the resources in such a way that the composition of available resources is better prepared to respond to the sub-hourly variations of the load and renewable resources in real-time operation. Therefore, this project paves the way to sustainable, reliable, and economic integration of renewable energy resources in power system, supporting the progress towards reaching the national targets on energy independence. Even if the proposed models offers a radically different point of view as compared to existing models, it does not alter fundamentally the architecture of power systems operations, nor the complexity of the scheduling problem, so the integration of this project in power systems is extremely practical.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2021

Multilane Automated Driving With Optimal Control and Mixed-Integer Programming

Road vehicle lane changes often initiate traffic disturbances and can therefore impact road networks’ energy and time efficiency. Furthermore, unexpected changes in traffic conditions may also render lane changes counterproductive for the lane-changing vehicle. Vehicle-to-vehicle connectivity combined with anticipative control could address these challenges via improved lane change decisions by automated vehicles. In a move toward this objective, receding horizon control cast as a mixed-integer quadratic program is used to plan lane changing and acceleration in a coupled optimization. A long-term pacing module, based on Pontryagin’s minimum principle from optimal control theory, sets terminal and input references for receding horizon control to target a user’s expected travel time. To remove nonlinear vehicle dynamics from the receding horizon controller, lane change commands are passed to a pure pursuit steering module whose response is approximated by a second-order linear model. Here, comparison against a rule-based reactive algorithm in arterial and highway scenarios shows an 8.9%–13.7% reduction in energy consumption and a 5.2%–10.3% reduction in the travel time, along with navigational improvements.

33 ADVANCED PROPULSION SYSTEMS↗
DOE OSTI2018

Mixed-integer programming models for optimal constellation scheduling given cloud cover uncertainty

We introduce the problem of scheduling observations on a constellation of remote sensors, to maximize the aggregate quality of the collections obtained. While automated tools exist to schedule remote sensors, they are often based on heuristic scheduling techniques, which typically fail to provide bounds on the quality of the resultant schedules. To address this issue, we first introduce a novel deterministic mixed-integer programming (MIP) model for scheduling a constellation of one to n satellites, which relies on extensive pre-computations associated with orbital propagators and sensor collection simulators to mitigate model size and complexity. Our MIP model captures realistic and complex constellation-target geometries, with solutions providing optimality guarantees. We then extend our base deterministic MIP model to obtain two-stage and three-stage stochastic MIP models that proactively schedule to maximize expected collection quality across a set of scenarios representing cloud cover uncertainty. Our experimental conclusions on instances of one and two satellites demonstrate that our stochastic MIP models yield significantly improved collection quality relative to our base deterministic MIP model. We further demonstrate that commercial off-the-shelf MIP solvers can produce provably optimal or near-optimal schedules from these models in time frames suitable for sensor operations.

47 OTHER INSTRUMENTATION↗
DOE OSTI2016

A superstructure-based framework for bio-separation network synthesis

Modern biotechnologies enable the production of chemicals using engineered microorganisms. However, the cost of downstream recovery and purification steps is high, which means that the feasibility of bio-based chemicals production depends heavily on the synthesis of cost-effective separation networks. To this end, we develop a superstructure-based framework for bio-separation network synthesis. Based on general separation principles and insights obtained from industrial processes for specific products, we first identify four separation stages: cell treatment, product phase isolation, concentration and purification, and refinement. For each stage, we systematically implement a set of connectivity rules to develop stage-superstructures, all of which are then integrated to generate a general superstructure that accounts for all types of chemicals that can be produced using microorganisms. Here we further develop a superstructure reduction method to solve specific instances, based on product attributes, technology availability, case-specific considerations, and final product stream specifications. A general optimization model, including short-cut models for all technologies, is formulated. The proposed framework enables preliminary synthesis and analysis of bio-separation networks, and thus estimation of separation costs.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
NASA NTRS1987

A satellite system synthesis model for orbital arc allotment optimization

A mixed integer programming formulation of a satellite system synthesis problem if presented, which is referred to as the arc allotment problem (AAP). Each satellite administration is to be allotted a weighted-length segment of the geostationary orbital arc within which its satellites may be positioned at any longitudes. The objective function maximizes the length of the unweighted arc segment allotted to every administration, subject to single-entry co-channel interference restrictions and constraints imposed by the visible arc for each administration. Useful relationships between special cases of AAP and another satellite synthesis problem are established. Solutions to two example problems are presented.

Reilly, Charles H.↗
DOE OSTI2019

A scalable mixed-integer decomposition approach for optimal power system restoration

The optimal restoration problem lies at the foundation of the evaluation and improvement of resilience in power systems. In this paper we present a scalable decomposition algorithm, based on the integer L-shaped method, for solving this problem for realistic power systems. The algorithm works by partitioning the problem into a master problem and a slave problem. The master problem optimizes over energization sequences of generators, buses and branches, for the entire restoration horizon, while respecting stylized energization requirements. The slave problem verifies that there exist power flow solutions supporting the energization sequences proposed by the master problem, adding cuts to the master if an infeasible sequence is detected. The power flow verification is carried out using a piece-wise linear approximation of the power flow equations, capable of capturing overvoltage situations common in restoration procedures. We develop specialized binary cuts and hybrid Benders-binary cuts for separating infeasible islands, which are stronger than regular no-good cuts, as well as heuristics for obtaining feasible solutions. We present numerical results for the IEEE 39- and 118-bus systems, and for the Chilean 1500-bus grid, where our algorithm finds near-optimal restoration sequences orders of magnitude faster than the state-of-the-art, demonstrating the effectiveness of the proposed approach.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2019

SUSPECT: MINLP special structure detector for Pyomo

We present SUSPECT, an open source toolkit that symbolically analyzes mixed-integer nonlinear optimization problems formulated using the Python algebraic modeling library Pyomo. We present the data structures and algorithms used to implement SUSPECT. SUSPECT works on a directed acyclic graph representation of the optimization problem to perform: bounds tightening, bound propagation, monotonicity detection, and convexity detection. We show how the tree-walking rules in SUSPECT balance the need for lightweight computation with effective special structure detection. SUSPECT can be used as a standalone tool or as a Python library to be integrated in other tools or solvers. Here, we highlight the easy extensibility of SUSPECT with several recent convexity detection tricks from the literature. We also report experimental results on the MINLPLib 2 dataset.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Optimal Allocation of Static Var Compensator via Mixed Integer Conic Programming

Shunt FACTS devices, such as, a Static Var Compensator (SVC), are capable of providing local reactive power compensation. They are widely used in the network to reduce the real power loss and improve the voltage profile. This paper proposes a planning model based on mixed integer conic programming (MICP) to optimally allocate SVCs in the transmission network considering load uncertainty. The load uncertainties are represented by a number of scenarios. Reformulation and linearization techniques are utilized to transform the original non-convex model into a convex second order cone programming (SOCP) model. Numerical case studies based on the IEEE 30-bus system demonstrate the effectiveness of the proposed planning model.

Zhang, Xiaohu↗
DOE OSTI2022

Protection settings optimizer

SAND2023-06672O The Protection Settings Optimizer (PSO) uses system and fault data as inputs to formulate the problem of calculating relay settings as a mixed integer, nonlinear optimization problem (MINLP). The MINLP is solved using a genetic algorithm-based optimizer that attempts to find settings to reduce the relay operating times. The PSO protects the power system by using the steady-state fault voltages and currents, which then calculates the optimal device setting to protect the power system. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Patel, Trupal↗
DOE OSTI2017

On Distributed PV Hosting Capacity Estimation, Sensitivity Study, and Improvement

This paper first studies the estimated distributed PV hosting capacities of seventeen utility distribution feeders using the Monte Carlo simulation based stochastic analysis, and then analyzes the sensitivity of PV hosting capacity to both feeder and photovoltaic system characteristics. Furthermore, an active distribution network management approach is proposed to maximize PV hosting capacity by optimally switching capacitors, adjusting voltage regulator taps, managing controllable branch switches and controlling smart PV inverters. The approach is formulated as a mixed-integer nonlinear optimization problem and a genetic algorithm is developed to obtain the solution. Multiple simulation cases are studied and the effectiveness of the proposed approach on increasing PV hosting capacity is demonstrated.

14 SOLAR ENERGY↗
DOE OSTI2022

Sequential linear integer programming for integer optimal control with total variation regularization

We propose a trust-region method that solves a sequence of linear integer programs to tackle integer optimal control problems regularized with a total variation penalty. The total variation penalty implies that the considered integer control problems admit minimizers. We introduce a local optimality concept for the problem, which arises from the infinite-dimensional perspective. In the case of a one-dimensional domain of the control function, we prove convergence of the iterates produced by our algorithm to points that satisfy first-order stationarity conditions for local optimality. We demonstrate the theoretical findings on a computational example.

97 MATHEMATICS AND COMPUTING↗