# Browsing by Subject "Mathematical model"

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Item Excel model for electric markets : ERCOT(2016-05) Cuevas, Pedro Pablo; Dyer, James S.; Butler, John C.(Clinical associate professor); Hahn, JoeShow more The effects of changing regulatory and fuel-cost environments have far reaching implications on the ability of electric markets to plan and provide cheap, clean, and reliable electric grids. The current state of the art tools for modeling the regulations and fuel prices requires days to process and access to these tools is also held by a small number licensed users that must also have the training and technical ability to run the model, which limits the study of planning and electricity market design.. This thesis presents an Excel model that simulates the operations of ERCOT over the next fifteen years. Tradeoffs between accuracy, run time, cost, and model complexity will be discussed. The advantages of this model are speed and accessibility, which will allow more users to understand the major implications of policy discussions and scenarios without needing a commercial tool. The model predicts the fuel mix and average market price for 2014 with less than a 1% and 2% error respectively. For 2015, the model predicts the fuel mix with less than a 5% error. Using the current trends assumptions, the model predicts that by 2030 the energy mix will undergo significant changes. Coal generation will drop from 28% to 21%, while gas generation will decline from 48% to 46%. Renewable generation will increase with wind going from 12% to 17% and solar from 0% to 7%. The model also predicts that a carbon tax between $20 and $60 per short ton of CO2, could rise the operational and capital costs of ERCOT in present value terms until 2030 from $75 billion to $218 billion. Finally the model forecasts that the reserve margin in ERCOT will not reach the target of 13.75% in 2020 and that renewable energy addition does not affect this indicator. Even more, the reserve margin is increased when solar energy enters the market.Show more Item Modeling mitigation strategies for infectious disease pandemics in US cities(2021-03-31) Wang, Xutong, Ph. D.; Meyers, Lauren Ancel; Wilke, Claus; Hillis, David; Scott, JamesShow more The novel coronavirus SARS-CoV-2 emerged in 2019. It spread rapidly around the globe and was declared a pandemic by the World Health Organization on March 11, 2020. To mitigate the unprecedented threat of COVID-19, public health agencies, researchers and pharmaceutical companies launched massive efforts to develop effective vaccines and medical treatments. To buy time and save lives, policymakers issued social distancing, mass gathering and face mask orders while ramping up programs for SARS-CoV-2 testing, contact tracing and isolation. In this dissertation, I developed data-driven mathematical models to evaluate the impacts of different intervention strategies on SARS-CoV-2 disease dynamics at a city-level. Chapter 1 provides a brief introduction and the historical context for the four studies, each of which modeled transmission dynamics and mitigation strategies at different stages of the pandemic. In Chapter 2, I estimated the impacts of various school closure and social distancing measures on local healthcare demands. This analysis supported pandemic decision making in Austin, Texas, including the local shelter-in-place order enacted on March 24, 2020. In Chapter 3, I modeled the impacts of relaxing social distancing orders following the initial stay-home orders across the US. For this study, I calibrated the transmission rate in my model based on hospitalization data from the city of Austin and estimated the impacts of cocooning (i.e., sheltering) high-risk individuals as measures relaxed. In Chapter 4, I examined the potential impacts of testing, contact tracing, and case isolation on disease transmission. Using data from a test-trace program jointly administered by Austin Public Health and the University of Texas, I evaluated the relative importance of testing and tracing speed and capacity and found that available resources are insufficient to slow transmission, particularly during times when the virus is spreading rapidly. In Chapter 5, I developed a model to compare SARS-CoV-2 vaccination strategies, that incorporates published efficacy estimates for the Moderna and Pfizer-BioNTech vaccines that received Emergency Use Approval in the US in late 2020. I evaluated the impacts of the timing of the vaccine rollout, the prioritization of vaccines to risk groups and age groups, rates of uptake, and single-dose versus two-dose strategies. Chapter 6 discusses the public health impacts, limitations, and lessons learned from the studies presented in Chapters 2-5. The research presented in this dissertation elucidated SARS-CoV-2 transmission dynamics and effective mitigation strategies at multiple stages of the COVID-19 pandemic and supported decision making by Austin, Texas and the US Centers for Disease Control and Prevention.Show more Item Sensitivity of AVO reflectivity to fluid properties in porous media(2004-05) Stine, Jason Andrew; Tatham, R. H. (Robert H.), 1943-Show more The Zoeppritz equations used in a typical reflection amplitude versus source-receiver offset (AVO) study to calculate the reflection and transmission coefficients do not directly consider the fluids filling the pore space in a porous solid medium. Although they account for the effects on the density and P-wave and S wave velocities in porous solids, these equations neglect the movements of fluids with respect to the porous framework. In doing so, the effects of the permeability and viscosity of the fluids during flow are ignored. These properties may affect the energy reflected and transmitted at a boundary; therefore, they must be accounted for to give an accurate wave propagation model. Biot theory considers the propagation of elastic waves in a porous elastic solid saturated by a viscous fluid. This theory accounts for the motion of fluids in the interconnected voids of a porous solid, assuming Darcian fluid flow. Biot theory accounts for the propagation of three waves, one rotational (shear) wave and two dilational waves (P-wave and slow wave). Reflection and transmission coefficients are calculated including Biot theory, showing potentially observable differences from the coefficients calculated using the Zoeppritz equations, for different physical situations. The sensitivity of the reflection coefficients to different physical parameters is examined. The goal is to evaluate how the reflection coefficients change as individual parameters, such as viscosity or permeability, are varied, and which parameters affect the reflection coefficients the most. If the reflection coefficient does not change as a parameter is varied, there is no sensitivity to that parameter and information about that parameter cannot be extracted from the data. The sensitivity analysis is complimented by calculating partial derivatives of the expressions for the reflection coefficients with respect to individual parameters, particularly fluid parameters. With this approach, large values of the partial derivative imply large changes in reflection coefficients with respect to a physical parameter indicate the most sensitivity to that parameter in the reflection coefficient. In Biot theory, the solid properties dominate over those of the fluids alone. The fluid properties only impact the reflection coefficients in a significant manner when there is a small contrast in the solid properties across a boundary. If the contrast in solid properties is too large, any effects caused by the fluid properties are insignificant compared to the solid effects. The three shale over sandstone models have too large of a contrast in solid properties to see fluid effects. Conversely, the six models of fluid boundaries within a reservoir sand all have little to no contrast in solid properties, so the fluid effects are evident. For gas-water interfaces, the observable changes in the P-P reflectivity are estimated to be as large as 5% for a 1% change in permeability and 15% for a 1% change in viscosity. When the above criteria for observing the fluid effects are met, the P-wave has sensitivity to viscosity, sensitivity, and porosity, with the reflection coefficients giving the most sensitivity to changes in the fluid viscosity. The apparent sensitivity to porosity is mostly a response to the density change caused by the change in porosity, rather than direct effects of the porosity. Theoretical AVO reflection coefficient curves based on Biot theory are inverted using two and three term AVO inversions based on approximations of Zoeppritz reflectivity. There is significant error in the parameters extracted by the inversion for both the two- and three-term AVO inversions. The three-term Aki and Richards (2002) inversion produces inaccurate values of the physical parameters across the boundary. Standard AVO inversion algorithms based on Zoeppritz reflectivity have problems accurately calculating parameters for a porous medium where fluids can move. An intercept and gradient interpretation algorithm based on Biot theory is desirable to accurately extract physical properties in porous media. A second formulation for reflectivity in a porous elastic solid is examined. In this study the theory developed by de la Cruz and Spanos (1985) is modified from their high viscosity limits, to fit more common lighter-oil viscosity regimes. The equations of motion and boundary conditions developed as part of the Spanos theory are adapted for this application. The reflectivity problem is simplified to an eigenvalue problem, based on a number of assumptions. De la Cruz, Hube, and Spanos (1992) published their computed values of reflection coefficients for high viscosity fluids. However, the complexity of this theory makes it impractical, in this study, to follow through to calculation of reflection coefficients in a porous elastic solidShow more Item Thermal lensing in ocular media(2009-05) Vincelette, Rebecca Lee; Welch, Ashley J., 1933-; Rockwell, Benjamin A.Show more This research was a collaborative effort between the Air Force Research Laboratory (AFRL) and the University of Texas to examine the laser-tissue interaction of thermal lensing induced by continuous-wave, CW, near-infrared, NIR, laser radiation in the eye and its influence on the formation of a retinal lesion from said radiation. CW NIR laser radiation can lead to a thermal lesion induced on the retina given sufficient power and exposure duration as related to three basic parameters; the percent of transmitted energy to, the optical absorption of, and the size of the laser-beam created at the retina. Thermal lensing is a well-known phenomenon arising from the optical absorption, and subsequent temperature rise, along the path of the propagating beam through a medium. Thermal lensing causes the laser-beam profile delivered to the retina to be time dependent. Analysis of a dual-beam, multidimensional, high-frame rate, confocal imaging system in an artificial eye determined the rate of thermal lensing in aqueous media exposed to 1110, 1130, 1150 and 1318-nm wavelengths was related to the power density created along the optical axis and linear absorption coefficient of the medium. An adaptive optics imaging system was used to record the aberrations induced by the thermal lens at the retina in an artificial eye during steady-state. Though the laser-beam profiles changed over the exposure time, the CW NIR retinal damage thresholds between 1110-1319-nm were determined to follow conventional fitting algorithms which neglected thermal lensing. A first-order mathematical model of thermal lensing was developed by conjoining an ABCD beam propagation method, Beer's law of attenuation, and a solution to the heat-equation with respect to radial diffusion. The model predicted that thermal lensing would be strongest for small (< 4-mm) 1/e² laser-beam diameters input at the corneal plane and weakly transmitted wavelengths where less than 5% of the energy is delivered to the retina. The model predicted thermal lensing would cause the retinal damage threshold for wavelengths above 1300-nm to increase with decreasing beam-diameters delivered to the corneal plane, a behavior which was opposite of equivalent conditions simulated without thermal lensing.Show more