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DOE OSTI2016

Study of a mixed dispersal population dynamics model

In this study, we consider a mixed dispersal model with periodic and Dirichlet boundary conditions and its corresponding linear eigenvalue problem. This model describes the time evolution of a population which disperses both locally and non-locally. We investigate how long time dynamics depend on the parameter values. Furthermore, we study the minimization of the principal eigenvalue under the constraints that the resource function is bounded from above and below, and with a fixed total integral. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for the species to die out more slowly or survive more easily. Our numerical simulations indicate that the optimal favorable region tends to be a simply-connected domain. Numerous results are shown to demonstrate various scenarios of optimal favorable regions for periodic and Dirichlet boundary conditions.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2014

Topography-based Flood Planning and Optimization Capability Development Report

Globally, water-related disasters are among the most frequent and costly natural hazards. Flooding inflicts catastrophic damage on critical infrastructure and population, resulting in substantial economic and social costs. NISAC is developing LeveeSim, a suite of nonlinear and network optimization models, to predict optimal barrier placement to protect critical regions and infrastructure during flood events. LeveeSim currently includes a high-performance flood model to simulate overland flow, as well as a network optimization model to predict optimal barrier placement during a flood event. The LeveeSim suite models the effects of flooding in predefined regions. By manipulating a domain’s underlying topography, developers altered flood propagation to reduce detrimental effects in areas of interest. This numerical altering of a domain’s topography is analogous to building levees, placing sandbags, etc. To induce optimal changes in topography, NISAC used a novel application of an optimization algorithm to minimize flooding effects in regions of interest. To develop LeveeSim, NISAC constructed and coupled hydrodynamic and optimization algorithms. NISAC first implemented its existing flood modeling software to use massively parallel graphics processing units (GPUs), which allowed for the simulation of larger domains and longer timescales. NISAC then implemented a network optimization model to predict optimal barrier placement based on output from flood simulations. As proof of concept, NISAC developed five simple test scenarios, and optimized topographic solutions were compared with intuitive solutions. Finally, as an early validation example, barrier placement was optimized to protect an arbitrary region in a simulation of the historic Taum Sauk dam breach.

42 ENGINEERING↗

DOE OSTI2018

Probabilistic Wind Power Ramp Forecasting Based on a Scenario Generation Method

Wind power ramps (WPRs) are particularly important in the management and dispatch of wind power and currently drawing the attention of balancing authorities. With the aim to reduce the impact of WPRs for power system operations, this paper develops a probabilistic ramp forecasting method based on a large number of simulated scenarios. An ensemble machine learning technique is first adopted to forecast the basic wind power forecasting scenario and calculate the historical forecasting errors. A continuous Gaussian mixture model (GMM) is used to fit the probability distribution function (PDF) of forecasting errors. The cumulative distribution function (CDF) is analytically deduced. The inverse transform method based on Monte Carlo sampling and the CDF is used to generate a massive number of forecasting error scenarios. An optimized swinging door algorithm is adopted to extract all the WPRs from the complete set of wind power forecasting scenarios. The probabilistic forecasting results of ramp duration and start-time are generated based on all scenarios. Numerical simulations on publicly available wind power data show that within a predefined tolerance level, the developed probabilistic wind power ramp forecasting method is able to predict WPRs with a high level of sharpness and accuracy.

17 WIND ENERGY↗

DOE OSTI2017

A Data-Driven Methodology for Probabilistic Wind Power Ramp Forecasting

With increasing wind penetration, wind power ramps (WPRs) are currently drawing great attention to balancing authorities, since these wind ramps largely affect power system operations. To help better manage and dispatch the wind power, this paper develops a data-driven probabilistic WPR forecasting (p-WPRF) method based on a large number of simulated scenarios. A machine learning technique is first adopted to forecast the basic wind power forecasting scenario and produce the historical forecasting errors. To accurately model the distribution of wind power forecasting errors, a generalized Gaussian mixture model is developed and the cumulative distribution function (CDF) is also analytically deduced. The inverse transform method based on the CDF is used to generate a large number of forecasting error scenarios. An optimized swinging door algorithm is adopted to extract all the WPRs from the complete set of wind power forecasting scenarios. The p-WPRF is generated based on all generated scenarios under different weather and time conditions. Numerical simulations on publicly available wind power data show that the developed p-WPRF method can predict WPRs with a high level of reliability and accuracy.

17 WIND ENERGY↗

DOE OSTI2017

Probabilistic Wind Power Ramp Forecasting Based on a Scenario Generation Method: Preprint

Wind power ramps (WPRs) are particularly important in the management and dispatch of wind power, and they are currently drawing the attention of balancing authorities. With the aim to reduce the impact of WPRs for power system operations, this paper develops a probabilistic ramp forecasting method based on a large number of simulated scenarios. An ensemble machine learning technique is first adopted to forecast the basic wind power forecasting scenario and calculate the historical forecasting errors. A continuous Gaussian mixture model (GMM) is used to fit the probability distribution function (PDF) of forecasting errors. The cumulative distribution function (CDF) is analytically deduced. The inverse transform method based on Monte Carlo sampling and the CDF is used to generate a massive number of forecasting error scenarios. An optimized swinging door algorithm is adopted to extract all the WPRs from the complete set of wind power forecasting scenarios. The probabilistic forecasting results of ramp duration and start time are generated based on all scenarios. Numerical simulations on publicly available wind power data show that within a predefined tolerance level, the developed probabilistic wind power ramp forecasting method is able to predict WPRs with a high level of sharpness and accuracy.

17 WIND ENERGY↗

DOE OSTI2025

Economic Storage Size Optimization for Electric Vehicle Extreme-Fast Charging Stations

En-route charging infrastructure for electric vehicles is critical to support transportation needs. These charging stations are likely to have high loads and especially sharp peak loads given fast charging capabilities needed to meet transportation schedules. In order to reduce both strain on distribution grid infrastructure and charging station operational costs, many stations are likely to employ behind the meter storage. This paper demonstrates a behind the meter storage sizing optimization that employs an open-source agent-based vehicle behavior model (BEAM) to determine the best sizing across many scenarios. This optimization and analysis is novel in that it examines how storage size impacts not only charging station cost and peak load, but also vehicle queue times. The optimization is also applied across a wide analysis region with sufficient diversity and numbers to provide novel statistical analysis of optimal sizes.

Aka, Julius↗

DOE OSTI2015

Reactive power planning under high penetration of wind energy using Benders decomposition

This study addresses the optimal allocation of reactive power volt-ampere reactive (VAR) sources under the paradigm of high penetration of wind energy. Reactive power planning (RPP) in this particular condition involves a high level of uncertainty because of wind power characteristic. To properly model wind generation uncertainty, a multi-scenario framework optimal power flow that considers the voltage stability constraint under the worst wind scenario and transmission N 1 contingency is developed. The objective of RPP in this study is to minimise the total cost including the VAR investment cost and the expected generation cost. Therefore RPP under this condition is modelled as a two-stage stochastic programming problem to optimise the VAR location and size in one stage, then to minimise the fuel cost in the other stage, and eventually, to find the global optimal RPP results iteratively. Benders decomposition is used to solve this model with an upper level problem (master problem) for VAR allocation optimisation and a lower problem (sub-problem) for generation cost minimisation. Impact of the potential reactive power support from doubly-fed induction generator (DFIG) is also analysed. Lastly, case studies on the IEEE 14-bus and 118-bus systems are provided to verify the proposed method.

17 WIND ENERGY↗

DOE OSTI2020

Sensor Placement Optimization Software Applied to Site-Scale Methane-Emissions Monitoring

Advances in sensor technology have increased our ability to monitor a wide range of environments. However, even as the cost of sensors decline, only a limited number of sensors can be installed at any given site. The physical placement of sensors, along with the sensor technology and operating conditions, can have a large impact on our ability to adequately monitor environmental change. This paper introduces a new open-source Python package, called Chama, that determines optimal sensor placement and technology to improve a sensor network’s detection capabilities. Additionally, the methods are demonstrated using site-specific methane emission scenarios that capture uncertainty in wind conditions and emission characteristics. Mixed-integer linear programming formulations are used to determine sensor locations and detection thresholds that maximize detection of the emission scenarios. The optimized sensor networks consistently increase the ability to detect leaks, as compared to sensors placed near each potential emission source or along the perimeter of the site.

47 OTHER INSTRUMENTATION↗

DOE OSTI2018

Sustainable Data Evolution Technology (SDET) for Power Grid Optimization (Final Report)

This project develops an innovative sustainable data evolution technology (SDET) to create open-access power grid datasets and facilitate updates to these datasets by the power grid community. The lack of open-access, realistic large-scale datasets significantly limits the ability of researchers, developers and ultimately end users to develop, benchmark and compare new methods and tools for optimizing the operation and planning of the grid, which leads to slow adoption by end users. Available open datasets include the IEEE sponsored 14-300 bus transmission test cases and distribution test feeders, the Edinburgh Power Systems Test Case Archive (39-2224 buses), and the Polish cases (2383-3375 buses) available with MATPOWER. These datasets serve certain purposes, but for power grid optimization, they are insufficient in one or more of these aspects: 1) they are too small; 2) they are incomplete in both models and scenarios for optimization purposes; and 3) they are static and are not keeping up with grid needs. The SDET technology delivers large-scale realistic datasets and data-creation tools capable of generating new datasets. We work closely with the Category-2 data repository project awarded by the same GRID DATA program. The generated datasets and data creation tools are compatible with and available through the data repository. The objective is to make this a sustained effort within and beyond the ARPA-E GRID DATA program so that the datasets can evolve over time and meet the current and future needs for power grid optimization and potentially other applications in power grid operation and planning. The novel and disruptive features of SDET differentiate itself from prior work in the following aspects: 1) SDET derives features and metrics for both transmission and distribution (T&D) systems by analyzing many public and private datasets provided by our industry partners National Rural Electric Cooperative Association (NRECA) and PJM Interconnection (PJM). 2) SDET develops data-creation tools and use these tools to generate large-scale open-access realistic datasets that comply with the metrics for both T&D systems. 3) SDET validates the created datasets using industry tools provided by our vendor partner GE Energy Solution. The data creation tools, as well as the created datasets, are integrated into the data repository (DR. POWER) funded by ARPA-E GRID DATA program in Category 2. Users can use the datasets and access these data creation tools to create and update datasets, thus datasets can evolve per user requirements and power grid development. The SDET is a novel and disruptive technology compared to current ad hoc and static dataset generation. It is expected the SDET technology will fundamentally enhance the development, assessment, and adoption of new methods and tools for power grid optimization and other applications, which in turn would significantly improve the reliability, resiliency, and efficiency of the power grid – a major ARPA-E mission objective.

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRS1989

GOES-I/M ascent maneuvers from transfer orbit to station

The Geostationary Operational Environmental Satellite (GOES)-I/M station acquisition sequence consists nominally of three in-plane/out-of-plane maneuvers at apogee on the line of relative nodes and a small in-plane maneuver at perigee. Existing software to determine maneuver attitude, ignition time, and burn duration required modification to optimize the out-of-plane parts and admit the noninertial, three-axis stabilized attitude. The Modified Multiple Impulse Station Acquisition Maneuver Planning Program (SENARIO2) was developed from its predecessor, SCENARIO, to optimize the out-of-plane components of the impulsive delta-V vectors. Additional new features include commputation of short term J sub 2 perturbations and output of all premaneuver and postmaneuver orbit elements, coarse maneuver attitudes, propellant usage, spacecraft antenna aspect angles, and ground station coverage. The output data are intended to be used in the launch window computation and by the maneuver targeting computation (General Maneuver (GMAN) Program) software. The maneuver targeting computation in GMAN was modified to admit the GOES-I/M maneuver attitude. Appropriate combinations of ignition time, burn duration, and attitude enable any reasonable target orbit to be achieved.

Abeyagunawardene, S.↗

DOE OSTI2014

TAS: 89 0227: TAS Recovery Act - Optimization and Control of Electric Power Systems: ARRA

The name SuperOPF is used to refer several projects, problem formulations and soft-ware tools intended to extend, improve and re-define some of the standard methods of optimizing electric power systems. Our work included applying primal-dual interior point methods to standard AC optimal power flow problems of large size, as well as extensions of this problem to include co-optimization of multiple scenarios. The original SuperOPF problem formulation was based on co-optimizing a base scenario along with multiple post-contingency scenarios, where all AC power flow models and constraints are enforced for each, to find optimal energy contracts, endogenously determined locational reserves and appropriate nodal energy prices for a single period optimal power flow problem with uncertainty. This led to example non-linear programming problems on the order of 1 million constraints and half a million variables. The second generation SuperOPF formulation extends this by adding multiple periods and multiple base scenarios per period. It also incorporates additional variables and constraints to model load following reserves, ramping costs, and storage resources. A third generation of the multi-period SuperOPF, adds both integer variables and a receding horizon framework in which the problem type is more challenging (mixed integer), the size is even larger, and it must be solved more frequently, pushing the limits of currently available algorithms and solvers. The consideration of transient stability constraints in optimal power flow (OPF) problems has become increasingly important in modern power systems. Transient stability constrained OPF (TSCOPF) is a nonlinear optimization problem subject to a set of algebraic and differential equations. Solving a TSCOPF problem can be challenging due to (i) the differential-equation constraints in an optimization problem, (ii) the lack of a true analytical expression for transient stability in OPF. To handle the dynamics in TSCOPF, the set of differential equations can be approximated or converted into equivalent algebraic equations before they are included in an OPF formulation. In Chapter 4, a rigorous evaluation of using a predefined and fixed threshold for rotor angles as a mean to determine transient stability of the system is developed. TSCOPF can be modeled as a large-scale nonlinear programming problem including the constraints of differential-algebraic equations (DAE). Solving a TSCOPF problem can be challenging due to (i) the differential-equation constraints in an optimization problem, (ii) the lack of a true analytical expression for transient stability constraint in OPF. Unfortunately, even the current best TSCOPF solvers still suffer from the curse of dimensionality and unacceptable computational time, especially for large-scale power systems with multiple contingencies. In chapter 5, thse issues will be addressed and a new method to incorporate the transient stability constraints will be presented.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2015

Constructing rigorous and broad biosurveillance networks for detecting emerging zoonotic outbreaks

Determining optimal surveillance networks for an emerging pathogen is difficult since it is not known beforehand what the characteristics of a pathogen will be or where it will emerge. The resources for surveillance of infectious diseases in animals and wildlife are often limited and mathematical modeling can play a supporting role in examining a wide range of scenarios of pathogen spread. We demonstrate how a hierarchy of mathematical and statistical tools can be used in surveillance planning help guide successful surveillance and mitigation policies for a wide range of zoonotic pathogens. The model forecasts can help clarify the complexities of potential scenarios, and optimize biosurveillance programs for rapidly detecting infectious diseases. Using the highly pathogenic zoonotic H5N1 avian influenza 2006-2007 epidemic in Nigeria as an example, we determined the risk for infection for localized areas in an outbreak and designed biosurveillance stations that are effective for different pathogen strains and a range of possible outbreak locations. We created a general multi-scale, multi-host stochastic SEIR epidemiological network model, with both short and long-range movement, to simulate the spread of an infectious disease through Nigerian human, poultry, backyard duck, and wild bird populations. We chose parameter ranges specific to avian influenza (but not to a particular strain) and used a Latin hypercube sample experimental design to investigate epidemic predictions in a thousand simulations. We ranked the risk of local regions by the number of times they became infected in the ensemble of simulations. These spatial statistics were then complied into a potential risk map of infection. Finally, we validated the results with a known outbreak, using spatial analysis of all the simulation runs to show the progression matched closely with the observed location of the farms infected in the 2006-2007 epidemic.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2016

3-cm fine structure masers: A unique signature of supermassive black hole formation via direct collapse in the early universe

The direct collapse black hole (DCBH) scenario describes the isothermal collapse of a pristine gas cloud directly into a massive, M{sub BH} = 10{sup 4}–10{sup 6}M{sub ⊙} black hole. In this paper we show that large H i column densities of primordial gas at T∼10{sup 4} K with low molecular abundance—which represent key aspects of the DCBH scenario—provide optimal conditions for the pumping of the 2p-level of atomic hydrogen by trapped Lyα photons. This Lyα pumping mechanism gives rise to an inverted level population of the 2s{sub 1/2}−2p{sub 3/2} transition, and therefore also gives rise to stimulated fine structure emission at λ=3.04 cm (rest-frame). We show that simplified models of the DCBH scenario amplify the CMB by up to a factor of ∼10{sup 5}, above which the maser saturates. Hyperfine splitting of the 3 cm transition gives rise to a characteristic broad (FWHM ∼ tens of MHz in the observers frame) asymmetric line profile. This signal subtends an angular scale of ∼1–10 mas, which translates to a flux of ∼0.3–3 μJy, which is detectable with ultra-deep surveys being planned with SKA1-MID. While challenging, as the signal is visible for a fraction of the collapse time of the cloud, the matching required physical conditions imply that a detection of the redshifted 3-cm emission line could provide direct evidence for the DCBH scenario.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2022

Simulation and Optimization of Volatile Fatty Acid Upgrading Strategies for Sustainable Transportation Fuel Production

Underutilized wet waste is a promising feedstock for production of carbon-neutral or carbon-negative liquid transportation fuels. Arrested methanogenesis of wet wastes by microbes, which produces mixtures of C2-C8 carboxylic acids (volatile fatty acids, VFAs), is practiced industrially at a pilot scale and the VFA products can be catalytically upgraded to molecules suitable for use as transportation fuels (alcohols and alkanes) via sequential ketonization and hydrogenation, both steps of which have been demonstrated with high (>90%) yield at the lab scale. We present in this work a simulation-based decision-making framework which evaluates upgrading strategies of VFAs to alcohol- and hydrocarbon-based neat and blended diesel, jet, and gasoline fuels. The processes utilize sequential upgrading steps (ketonization and hydrogenation) accompanied by one distillation step (either separating VFAs before ketonization or alcohols after ketone hydrogenation) to form two liquid streams: a light alcohol stream best suited as a light-duty fuel and a heavy alcohol or hydrocarbon stream best suited as a diesel or jet fuel. Suitability of the liquid products as transportation fuels and maximum blending levels with petrofuels comes from simulation of critical fuel properties (boiling point, flash point, lower heating value, viscosity, melting point, water solubility, and cetane/octane number). Catalytic upgrading steps and fuel property mixing models used by the simulation are experimentally validated. We demonstrate the flexibility of the decision-making algorithm by evaluating processing scenarios for experimentally-observed VFA profiles with varied carbon chain length distributions. Processing scenarios are optimized for either maximum total renewable carbon utilization or production of heavy-duty (jet or diesel) fuels. We evaluate the economic viability and CO2 emissions of each proposed upgrading scenario using techno-economic and life cycle analyses. The decision-making framework developed in this work can also be used to down-select promising strategies for upgrading of other bio-based feedstocks to sustainable fuels and chemicals.

aviation fuel↗

DOE OSTI2016

Final Technical Report: Sparse Grid Scenario Generation and Interior Algorithms for Stochastic Optimization in a Parallel Computing Environment

The support from this grant resulted in seven published papers and a technical report. Two papers are published in SIAM J. on Optimization [87, 88]; two papers are published in IEEE Transactions on Power Systems [77, 78]; one paper is published in Smart Grid [79]; one paper is published in Computational Optimization and Applications [44] and one in INFORMS J. on Computing [67]). The works in [44, 67, 87, 88] were funded primarily by this DOE grant. The applied papers in [77, 78, 79] were also supported through a subcontract from the Argonne National Lab. We start by presenting our main research results on the scenario generation problem in Sections 1–2. We present our algorithmic results on interior point methods for convex optimization problems in Section 3. We describe a new ‘central’ cutting surface algorithm developed for solving large scale convex programming problems (as is the case with our proposed research) with semi-infinite number of constraints in Section 4. In Sections 5–6 we present our work on two application problems of interest to DOE.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Integrated GW Farm ABM

This Data Repository includes data used for the integrated groundwater- farm ABM model, raw model output from scenario ensemble, and processed outputs that isolate the groundwater storage depletion outcomes for the 35,000 farm cells. Model Inputs: Farm ABM Inputs: This folder contains the input data used by the integrated groundwater - farm ABM modelling script (Python file) used for the high performance computing (HPC) experiments. The sub-folder "data inputs" contains all of the farm attribute data, while the three files in the folder have the hydrogeological data lookup table (NLDAS Cost Curve Attributes.csv), a lookup table (Theis well function table.csv) for the groundwater cost curve function, and the farm indexes and corresponding NLDAS ids for all of the cells run in this experiment (nldas farms subset final.csv). NLDAS Cost curve hydrogeological data: Hydrogeological data aggregated to 1/8 degree resolution and aligned with the NLDAS grid. Parameters include: water depth below ground surface [meters], subsurface porosity [unitless], aquifer depth from ground surface to aquifer bottom [meters], annual average recharge (USGS: mm, Doll: meters), and three different hydraulic conductivity (K) values (meters/day). The three K values represent the mean value from Gleeson et al. (2018), one standard deviation above the mean from Gleeson et al. (2018), and the de Graaf et al. 2020 modifications to certain lithologies. Additional information about these datasets and their processing are documented in the supplement to Yoon et al. 2025 (in review). Output: Raw outputs: This folder contains a .zip file that has model outputs for the entire scenario ensemble. There is one csv for each farm id, using the format "farm farmid cases.csv". The relationship between the farm id and NLDAS id is defined by the "nldas farms subset final.csv" located in the Farm ABM Inputs folder. Each csv has 625 rows, corresponding to 625 combinations of different scenario parameter values. Each row (scenario) represents the outcome of a 100 year simulation. Columns define scenario settings and summary statistics for each scenario. The first four columns define the scenario settings: "hydro ratio," "econ ratio," "K scenario," and "gamma scenario." The hydro and econ ratios are values passed to the modeling script that influence multipliers for other model parameters, as documented in the supplement to Yoon et al. 2025 (in review). The gamma multiplier is a coefficient multiplier applied to the baseline gamma values (values below 1 represent lower unobserved costs compared to baseline, values above 1 represent higher costs). The K scenario names represent K values of: "low": 0.5 m/d, "int 1": 2.5 m/d, "int 2": 10 m/d, "high": 50 m/d, and "gleeson": mean Gleeson K value. "Perc vol depleted" is the fraction of groundwater depleted at the end of the 100 simulation. Processed Output: Derived depletion outcomes from raw outputs: All of the individual csv files from the Raw outputs were aggregated into a single file that has the scenario settings and fraction depletion "Perc vol depleted" for every farm cell, for every scenario. The other two files define relationships between the farm id, NLDAS id, and local and major aquifer units, used for aquifer-level depletion analysis.

Agent based modeling↗

NASA NTRS2006

Solid Modeling of Crew Exploration Vehicle Structure Concepts for Mass Optimization

Parametric solid and surface models of the crew exploration vehicle (CEV) command module (CM) structure concepts are developed for rapid finite element analyses, structural sizing and estimation of optimal structural mass. The effects of the structural configuration and critical design parameters on the stress distribution are visualized, examined to arrive at an efficient design. The CM structural components consisted of the outer heat shield, inner pressurized crew cabin, ring bulkhead and spars. For this study only the internal cabin pressure load case is considered. Component stress, deflection, margins of safety and mass are used as design goodness criteria. The design scenario is explored by changing the component thickness parameters and materials until an acceptable design is achieved. Aluminum alloy, titanium alloy and an advanced composite material properties are considered for the stress analysis and the results are compared as a part of lessons learned and to build up a structural component sizing knowledge base for the future CEV technology support. This independent structural analysis and the design scenario based optimization process may also facilitate better CM structural definition and rapid prototyping.

Mukhopadhyay, Vivek↗

NASA NTRS1993

Optimal return-to-launchsite abort trajectories for an HL-20 Personnel Launch System vehicle

The Personnel Launch System (PLS) being studied by NASA is a system to complement the Space Shuttle and provide alternative access to space. The PLS consists of a manned spacecraft launched by an expendable launch vehicle (ELV). A candidate for the manned spacecraft is the HL-20 lifting body. In the event of an ELV malfunction during the initial portion of the ascent trajectory, the HL-20 will separate from the rocket and perform an unpowered return-to-launchsite (RTLS) abort. This paper describes an investigation of the RTLS abort scenario using optimal control theory. The objective of the abort trajectory is to maximize final altitude at a point near the runway. The assumption is then made that there exists a control history to steer the vehicle to any final altitude lower than the final optimal altitude. With this selection of cost function, and with this assumption, the feasibility of an RTLS abort at different times along the ascent trajectory can be determined. The method of differential inclusions, which allows the determination of optimal states and eliminates the need for determining the optimal controls, is used to determine the optimal trajectories.

Dutton, Kevin E.↗