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At least 343 records · Page 19

Final Report---Optimization Under Nonconvexity and Uncertainty: Algorithms and Software

the goal of this work was to develop new algorithmic techniques for solving large-scale numerical optimization problems, focusing on problems classes that have proven to be among the most challenging for practitioners: those involving uncertainty and those involving nonconvexity. This research advanced the state-of-the-art in solving mixed integer linear programs containing symmetry, mixed integer nonlinear programs, and stochastic optimization problems. The focus of the work done in the continuation was on Mixed Integer Nonlinear Programs (MINLP)s and Mixed Integer Linear Programs (MILP)s, especially those containing a great deal of symmetry.

97 MATHEMATICS AND COMPUTING↗

Connected Vehicle-Based Traffic Signal Coordination

This study presents a connected vehicles (CVs)-based traffic signal optimization framework for a coordinated arterial corridor. The signal optimization and coordination problem are first formulated in a centralized scheme as a mixed-integer nonlinear program (MINLP). The optimal phase durations and offsets are solved together by minimizing fuel consumption and travel time considering an individual vehicle’s trajectories. Due to the complexity of the model, we decompose the problem into two levels: an intersection level to optimize phase durations using dynamic programming (DP), and a corridor level to optimize the offsets of all intersections. In order to solve the two-level model, a prediction-based solution technique is developed. The proposed models are tested using traffic simulation under various scenarios. Compared with the traditional actuated signal timing and coordination plan, the signal timing plans generated by solving the MINLP and the two-level model can reasonably improve the signal control performance. When considering varies vehicle types under high demand levels, the proposed two-level model reduced the total system cost by 3.8% comparing to baseline actuated plan. MINLP reduced the system cost by 5.9%. It also suggested that coordination scheme was beneficial to corridors with relatively high demand levels. For intersections with major and minor street, coordination conducted for major street had little impacts on the vehicles at the minor street.

42 ENGINEERING↗

Introduction to Mixed Variable Optimization (MVO) [Slides]

Mixed Variable Optimization (MVO) is an optimization method that uses discrete (categorical) and continuous variables in its optimization process. This method of optimization is applicable to highly nonlinear problems where the information used in the optimization is of poor quality of has minor inaccuracies in the data sets.

97 MATHEMATICS AND COMPUTING↗

A Planning Problem Combining Calculus of Variations and Optimal Transport

We consider some variants of the classical optimal transport where not only one optimizes over couplings between some variables x and y but also over some control variables governing the evolutions of these variables with time. Such a situation is motivated by an assignment problem of tasks with workers whose characteristics can evolve with time (and be controlled). We distinguish between the coupled and decoupled case. The coupled case is a standard optimal transport with the value of some optimal control problem as cost. The decoupled case is more involved since it is nonlinear in the transport plan.

97 MATHEMATICS AND COMPUTING↗

An Efficient Parallel-in-Time Method for Optimization with Parabolic PDEs

To solve optimization problems with parabolic PDE constraints, often methods working on the reduced objective functional are used. They are computationally expensive due to the necessity of solving both the state equation and a backward-in-time adjoint equation to evaluate the reduced gradient in each iteration of the optimization method. In this study, we investigate the use of the parallel-in-time method PFASST in the setting of PDE-constrained optimization. In order to develop an efficient fully time-parallel algorithm, we discuss different options for applying PFASST to adjoint gradient computation, including the possibility of doing PFASST iterations on both the state and the adjoint equations simultaneously. Here, we also explore the additional gains in efficiency from reusing information from previous optimization iterations when solving each equation. Numerical results for both a linear and a nonlinear reaction-diffusion optimal control problem demonstrate the parallel speedup and efficiency of different approaches.

97 MATHEMATICS AND COMPUTING↗

Double Emittance Exchanger as a Bunch Compressor for the MaRIE XFEL electron beam line at 1GeV

We demonstrate an alternative realization of a bunch compressor (specifically the second bunch compressor for the MaRIE XFEL beamline, 1GeV electron energy) using a double emittance exchanger (EEX) and a telescope in the transverse phase space.We compare our results with a traditional bunch compressor realized via chicane, taking into account the nonlinear dynamics, Coherent Synchrotron Radiation (CSR) and Space Charge (SC) effects. In particular, we use the Elegant code for tracking particles through the beam line and analyze the eigen-emittances evolution to separate the influence of the CSR/SC effects from the nonlinear dynamics effects. We optimize the scheme parameters to reach a desirable compression factor and minimize the emittance growth. We observe dominant CSR-effects in our scheme resulting in critical emittance growth and introduce alternative version of an emittance exchanger with a reduced number of bending magnets to minimize the impact of CSR effects.

43 PARTICLE ACCELERATORS↗

Study of Integrable and Quasi-Integrable Sextupole Lattice

In order to maximize beam lifetime in circular particle accelerators, the nonlinear beam optics are optimized to maximize the dynamic aperture of the beam. The dynamic aperture (DA), which is a 6-D phase space volume of stable trajectories, depends on the strength of the nonlinearities in the machine, and is calculated via particle tracking. Current DA optimization processes include multi-objective genetic algorithm optimizers, and relies on minimizing the magnitudes of resonance driving terms (RDT), which are calculated from the nonlinear contribution to the one-turn-map. The process of searching through the parameter space for an ideal combination that maximizes DA is computationally strenuous. By setting up the sextupole channel such that it is resembles a symplectic integrator of a smooth Hamiltonian, with only a few sextupoles we are able to closely reproduce phase space trajectories of a smooth Hamiltonian up to the hyperbolic point. No chaos and resonances are observed if phase advance per one sextupole magnet in the channel does not exceed ~0.12x2 pi. Therefore, an important property of the suggested approach is the intrinsic elimination of the resonances, and minimization of corresponding RDTs.

43 PARTICLE ACCELERATORS↗

Joint analysis of Dark Energy Survey Year 3 data and CMB lensing from SPT and P l a n c k . II. Cross-correlation measurements and cosmological constraints

Cross-correlations of galaxy positions and galaxy shears with maps of gravitational lensing of the cosmic microwave background (CMB) are sensitive to the distribution of large-scale structure in the Universe. Such cross-correlations are also expected to be immune to some of the systematic effects that complicate correlation measurements internal to galaxy surveys. We present measurements and modeling of the cross-correlations between galaxy positions and galaxy lensing measured in the first three years of data from the Dark Energy Survey with CMB lensing maps derived from a combination of data from the 2500 deg 2 SPT-SZ survey conducted with the South Pole Telescope and full-sky data from the Planck satellite. The CMB lensing maps used in this analysis have been constructed in a way that minimizes biases from the thermal Sunyaev Zel’dovich effect, making them well suited for cross-correlation studies. The total signal-to-noise of the cross-correlation measurements is 23.9 (25.7) when using a choice of angular scales optimized for a linear (nonlinear) galaxy bias model. Here we use the cross-correlation measurements to obtain constraints on cosmological parameters. For our fiducial galaxy sample, which consist of four bins of magnitude-selected galaxies, we find constraints of Ω m =0.272$^{+0.032}_{-0.052}$ and S 8 ≡σ8$\sqrt{Ω_m/0.3}$=0.736$^{+0.032}_{-0.028}$ (Ω m =0.245$^{+0.026}_{-0.044}$ and S 8 =0.734$^{+0.035}_{-0.028}$) when assuming linear (nonlinear) galaxy bias in our modeling. Considering only the cross-correlation of galaxy shear with CMB lensing, we find Ω m =0.270$^{+0.043}_{-0.061}$ and S 8 =0.740$^{+0.034}_{-0.029}$. Our constraints on S8 are consistent with recent cosmic shear measurements, but lower than the values preferred by primary CMB measurements from Planck.

79 ASTRONOMY AND ASTROPHYSICS↗

Network-Level Optimization for Unbalanced Power Distribution System: Approximation and Relaxation

The nonlinear programming (NLP) problem to solve distribution-level optimal power flow (D-OPF) poses convergence issues and does not scale well for unbalanced distribution systems. The existing scalable D-OPF algorithms either use approximations that are not valid for an unbalanced power distribution system, or apply relaxation techniques to the nonlinear power flow equations that do not guarantee a feasible power flow solution. In this paper, we propose scalable D-OPF algorithms that simultaneously achieve optimal and feasible solutions by solving multiple iterations of approximate, or relaxed, D-OPF subproblems of low complexity. The first algorithm is based on a successive linear approximation of the nonlinear power flow equations around the current operating point, where the D-OPF solution is obtained by solving multiple iterations of a linear programming (LP) problem. The second algorithm is based on the relaxation of the nonlinear power flow equations as conic constraints together with directional constraints, which achieves optimal and feasible solutions over multiple iterations of a second-order cone programming (SOCP) problem. Finally, it is demonstrated that the proposed algorithms are able to reach an optimal and feasible solution while significantly reducing the computation time as compared to an equivalent NLPD-OPF model for the same distribution system.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Assessing the Optimality of LinDist3Flow for Optimal Tap Selection of Step Voltage Regulators in Unbalanced Distribution Networks

The adoption of distributed energy resources such as photovoltaics (PVs) has increased dramatically during the previous decade. The increased penetration of PVs into distribution networks (DNs) can cause voltage fluctuations that have to be mitigated. One of the key utility assets employed to this end are step-voltage regulators (SVRs). It is desirable to include tap selection of SVRs in optimal power flow (OPF) routines, a task that turns out to be challenging because the resultant OPF problem is nonconvex with added complexities stemming from accurate SVR modeling. While several convex relaxations based on semi-definite programming (SDP) have been presented in the literature for optimal tap selection, SDP based schemes do not scale well and are challenging to implement in large-scale planning or operational frameworks. This paper deals with the optimal tap selection (OPTS) problem for wye-connected SVRs using linear approximations of power flow equations. Specifically, the LinDist3Flow model is adopted and the effective SVR ratio is assumed to be continuous- enabling the formulation of a problem called LinDist3Flow-OPTS, which amounts to a linear program. The scalability and optimality gap of LinDist3Flow-OPTS are evaluated with respect to existing SDP-based and nonlinear programming techniques for optimal tap selection in three standard feeders, namely, the IEEE 13-bus, 123-bus, and 8500-node DNs. For all DNs considered, LinDist3Flow-OPTS achieves an optimality gap of approximately 1% or less while significantly lowering the computational burden.

linear approximations↗

Assessing the Optimality of LinDist3Flow for Optimal Tap Selection of Step Voltage Regulators in Unbalanced Distribution Networks: Preprint

The adoption of distributed energy resources such as photovoltaics (PVs) has increased dramatically during the previous decade. The increased penetration of PVs into distribution networks (DNs) can cause voltage fluctuations that have to be mitigated. One of the key utility assets employed to this end are step-voltage regulators (SVRs). It is desirable to include tap selection of SVRs in optimal power flow (OPF) routines, a task that turns out to be challenging because the resultant OPF problem is nonconvex with added complexities stemming from accurate SVR modeling. While several convex relaxations based on semi-definite programming (SDP) have been presented in the literature for optimal tap selection, SDP based schemes do not scale well and are challenging to implement in large-scale planning or operational frameworks. This paper deals with the optimal tap selection (OPTS) problem for wye-connected SVRs using linear approximations of power flow equations. Specifically, the LinDist3Flow model is adopted and the effective SVR ratio is assumed to be continuous–enabling the formulation of a problem called LinDist3Flow-OPTS, which amounts to a linear program. The scalability and optimality gap of LinDist3Flow-OPTS are evaluated with respect to existing SDP-based and nonlinear programming techniques for optimal tap selection in three standard feeders, namely, the IEEE 13-bus, 123-bus, and 8500-node DNs. For all DNs considered, LinDist3Flow-OPTS achieves an optimality gap of approximately 1% or less while significantly lowering the computational burden.

linear approximations↗

Modeling the AC Power Flow Equations with Optimally Compact Neural Networks: Application to Unit Commitment

Nonlinear power flow constraints render a variety of power system optimization problems computationally intractable. Emerging research shows, however, that the nonlinear AC power flow equations can be successfully modeled using neural networks. These neural networks can be exactly transformed into mixed integer linear programs and embedded inside challenging optimization problems, thus replacing nonlinearities that are intractable for many applications with tractable piecewise linear approximations. Such approaches, though, suffer from an explosion of the number of binary variables needed to represent the neural network. Accordingly, this paper develops a technique for training an "optimally compact'' neural network, i.e., one that can represent the power flow equations with a sufficiently high degree of accuracy while still maintaining a tractable number of binary variables. We demonstrate the use of this neural network as an approximator of the nonlinear power flow equations by embedding it in the AC unit commitment problem, transforming the problem from a mixed integer nonlinear program into a more manageable mixed integer linear program. We use the 14-, 57-, and 89-bus networks as test cases and compare the AC-feasibility of commitment decisions resulting from the neural network, DC, and linearized power flow approximations. Our results show that the neural network model outperforms both the DC and linearized power flow approximations when embedded in the unit commitment problem. The neural network formulation most often selects a feasible unit commitment schedule, and furthermore, it only s

AC power flow↗

Global optimization of multicomponent distillation configurations: Global minimization of total cost for multicomponent mixture separations

We introduce a global optimization framework for determining the minimum cost required to distill any ideal or near-ideal multicomponent mixture into its individual constituents using a sequence of columns. This new framework extends the Global Minimization Algorithm (GMA) previously introduced by Nallasivam et al. (2016); and we refer to the new framework as the Global Minimization Algorithm for Cost (GMAC). GMAC guarantees global optimality by formulating a nonlinear program (NLP) for each and every distillation configuration in the search space and solving it using global optimization solvers. In conclusion, the case study presented in this work not only demonstrates the need for developing such an algorithm, but also shows the flexibility and effectiveness of GMAC, which enables process engineers to design and retrofit energy efficient and cost-effective distillation configurations.

42 ENGINEERING↗

OPTIMIZATION FRAMEWORK FOR A RADIO FREQUENCY GUN BASED INJECTOR

Linear accelerator based light sources are used to produce coherent x-ray beams with unprecedented peak intensity. In these devices, the key parameters of the photon beam such as brilliance and coherence are directly dependent on the electron beam parameters. This leads to stringent beam quality requirements for the electron beam source. Radio frequency (RF) guns are used in such light sources since they accelerate electrons to relativistic energies over a very short distance, thus minimizing the beam quality degradation due to space charge effects within the particle bunch. Designing such sources including optimization of its beam parameters is a complex process where one needs to meet many requirements simultaneously. It is useful to have a tool to automate the design optimization in the context of the injector beam dynamics performance. Evolutionary and genetic algorithms are powerful tools to apply to nonlinear multi-objective optimization problems, and they have been successfully used in injector optimizations where the electric field profiles for the accelerating devices are fixed. Here the genetic algorithm based approach is extended to modify and optimize the electric field profile for an RF gun concurrently with the injector performance. Two field modification methods are used. This dissertation presents an overview of the optimization system and examples of its application to a state of the art RF gun. Results indicate improved injector performance is possible with unbalanced electric field profiles where the peak field in the cathode cell is larger than in subsequent cells.

Hofler, Alicia↗

Eigenfrequency constrained topology optimization of finite strain hyperelastic structures

This paper incorporates hyperelastic materials, nonlinear kinematics, and preloads in eigenfrequency constrained density–based topology optimization. The formulation allows for initial finite deformations and subsequent small harmonic oscillations. The optimization problem is solved by the method of moving asymptotes, and the gradients are calculated using the adjoint method. Both simple and degenerate eigenfrequencies are considered in the sensitivity analysis. A well-posed topology optimization problem is formulated by filtering the volume fraction field. Numerical issues associated with excessive distortion and spurious eigenmodes in void regions are reduced by removing low volume fraction elements. The optimization objective is to maximize stiffness subject to a lower bound on the fundamental eigenfrequency. Numerical examples show that the eigenfrequencies drastically change with the load magnitude, and that the optimization is able to produce designs with the desired fundamental eigenfrequency.

42 ENGINEERING↗

Holistic Portfolio Optimization using Directed Mutations

Genetic algorithms provide attractive options for performing nonlinear multi-objective combinatorial design optimization, and they have proven very useful for optimizing individual systems. However, conventional genetic algorithms fall short when performing holistic portfolio optimizations in which the decision variables also include the integer counts of multiple system types over multiple time periods. When objective functions are formulated as analytic functions, we can formally differentiate with respect to system counts and use the resulting gradient information to generate favorable mutations in the count variables. We apply several variations on this basic idea to an idealized hanging chain example to obtain >> 1000x speedups over conventional genetic algorithms in both single - and multi-objective cases. We develop a more complex example of a notional military portfolio that includes combinatorial design variables and dependency constraints between the design choices. In this case, our initial results are mixed, but many variations are still open to further research.

97 MATHEMATICS AND COMPUTING↗