Knowing the uncertainty of a calculated result is always important, but especially so when performing calculations for safety analysis. A traditional way of propagating the uncertainty of input parameters is Monte Carlo (MC) methods. A quicker alternative to MC, especially useful when computations are heavy, is Deterministic Sampling (DS). DS works by hand-picking a small set of samples, rather than randomizing a large set as in MC methods. The samples and its corresponding weights are chosen to represent the uncertainty one wants to propagate by encoding the first few statistical moments of the parameters' distributions. Finding a suitable ensemble for DS in not easy, however. Given a large enough set of samples, one can always calculate weights to encode the first couple of moments, but there is good reason to want an ensemble with only positive weights. How to choose the ensemble for DS so that all weights are positive is the problem investigated in this project. Several methods for generating such ensembles have been derived, and an algorithm for calculating weights while forcing them to be positive has been found. The methods and generated ensembles have been tested for use in uncertainty propagation in many different cases and the ensemble sizes have been compared. In general, encoding two or four moments in an ensemble seems to be enough to get a good result for the propagated mean value and standard deviation. Regarding size, the most favorable case is when the parameters are independent and have symmetrical distributions. In short, DS can work as a quicker alternative to MC methods in uncertainty propagation as well as in other applications.

One of the biggest threats to the power systems as critical infrastructures is large-scale blackouts resulting from cascading failures (CF) in the grid. The ongoing shift in energy portfolio due to ever-increasing penetration of renewable energy sources (RES) may drive the electric grid closer to its operational limits and introduce a large amount of uncertainty coming from their stochastic nature. One worrisome change is the increase in CFs. The CF simulation models in the literature do not allow consideration of RES penetration in studying the grid vulnerability. In this dissertation, we have developed tools and models to evaluate the impact of RE penetration on grid vulnerability to CF. We modeled uncertainty injected from different sources by analyzing actual high-resolution data from North American utilities. Next, we proposed two CF simulation models based on simplified DC power flow and full AC power flow to investigate system behavior under different operating conditions. Simulations show a dramatic improvement in the line flow uncertainty estimation based on the proposed model compared to the simplified DC OPF model. Furthermore, realistic assumptions on the integration of RE resources have been made to enhance our simulation technique. The proposed model is benchmarked against the historical blackout data and widely used models in the literature showing similar statistical patterns of blackout size.

Submitted by JÚLIO HEBER SILVA (julioheber@yahoo.com.br) on 2017-08-17T17:45:38Z No. of bitstreams: 2 Dissertação - Ademir Alves de Brito Júnior - 2016.pdf: 4581696 bytes, checksum: fb086c82c7a1661645e7f2006d2ae319 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2017-08-18T12:03:26Z (GMT) No. of bitstreams: 2 Dissertação - Ademir Alves de Brito Júnior - 2016.pdf: 4581696 bytes, checksum: fb086c82c7a1661645e7f2006d2ae319 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2017-08-18T12:03:26Z (GMT). No. of bitstreams: 2 Dissertação - Ademir Alves de Brito Júnior - 2016.pdf: 4581696 bytes, checksum: fb086c82c7a1661645e7f2006d2ae319 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-05-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, a numerical version of Unscented Transform was developed. In the developed approach, any probability distributions can be mapped by means of linear or non-linear functions, thus allowing fast acquisition of the probability distributions of the outputs/ simulation model responses, or more specifically, the evaluation of the uncertainty model. For practical purposes of distribution mapping, the computational cost is considerably lower than that demanded by the Monte Carlo method, which is based on a massive random sampling, thus presenting high computational cost. The application in Biomechanics problems shows the efficiency of the proposed method. / Neste trabalho, foi desenvolvida uma versão numérica da Transformada da Incerteza (expressão utilizada para denominar a Unscented Transform). Na abordagem elaborada, quaisquer distribuições de probabilidade podem ser mapeadas por meio de funções lineares ou não-lineares, permitindo assim a obtenção ágil das distribuições de probabilidade das saídas/respostas do modelo de simulação ou, mais especificamente, do modelo de avaliação de incertezas. Para propósitos práticos de mapeamento de distribuições, o custo computacional se mostra consideravelmente menor que aquele demandado pelo método de Monte Carlo, o qual é baseado em amostragem aleatória massiva, apresentando assim alto custo computacional. A aplicação em problemas de Biomecânica como a avaliação mecânica do osso humano e avaliação de incertezas da marcha humana por meio da dinâmica inversa, mostra a eficiência do método proposto em vista de outros métodos conhecidos como o de Monte Carlo.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2016-10-17T20:30:31Z No. of bitstreams: 2 Dissertação - Ademir Alves de Brito Júnior - 2016.pdf: 4972595 bytes, checksum: 78d986800cafd8a0bf84801e35391b16 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Approved for entry into archive by Erika Demachki (erikademachki@gmail.com) on 2016-10-18T17:16:43Z (GMT) No. of bitstreams: 2 Dissertação - Ademir Alves de Brito Júnior - 2016.pdf: 4972595 bytes, checksum: 78d986800cafd8a0bf84801e35391b16 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) / Made available in DSpace on 2016-10-18T17:16:43Z (GMT). No. of bitstreams: 2 Dissertação - Ademir Alves de Brito Júnior - 2016.pdf: 4972595 bytes, checksum: 78d986800cafd8a0bf84801e35391b16 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-05-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, a numerical version of Unscented Transform was developed. In the developed approach, any probability distributions can be mapped by means of linear or non-linear functions, thus allowing fast acquisition of the probability distributions of the outputs/ simulation model responses, or more specifically, the evaluation of the uncertainty model. For practical purposes of distribution mapping, the computational cost is considerably lower than that demanded by the Monte Carlo method, which is based on a massive random sampling, thus presenting high computational cost. The application in Biomechanics problems shows the efficiency of the proposed method. / Neste trabalho, foi desenvolvida uma versão numérica da Transformada da Incerteza (expressão utilizada para denominar a Unscented Transform). Na abordagem elaborada, quaisquer distribuições de probabilidade podem ser mapeadas por meio de funções lineares ou não-lineares, permitindo assim a obtenção ágil das distribuições de probabilidade das saídas/respostas do modelo de simulação ou, mais especificamente, do modelo de avaliação de incertezas. Para propósitos práticos de mapeamento de distribuições, o custo computacional se mostra consideravelmente menor que aquele demandado pelo método de Monte Carlo, o qual é baseado em amostragem aleatória massiva, apresentando assim alto custo computacional. A aplicação em problemas de Biomecânica como a avaliação mecânica do osso humano e avaliação de incertezas da marcha humana por meio da dinâmica inversa, mostra a eficiência do método proposto em vista de outros métodos conhecidos como o de Monte Carlo.

In this thesis, two novel methods for Inverse Uncertainty Quantification are benchmarked against the more established methods of Monte Carlo sampling of output parameters(MC) and Maximum Likelihood Estimation (MLE). Inverse Uncertainty Quantification (IUQ) is the process of how to best estimate the values of the input parameters in a simulation, and the uncertainty of said estimation, given a measurement of the output parameters. The two new methods are Deterministic Sampling (DS) and Weight Fixing (WF). Deterministic sampling uses a set of sampled points such that the set of points has the same statistic as the output. For each point, the corresponding point of the input is found to be able to calculate the statistics of the input. Weight fixing uses random samples from the rough region around the input to create a linear problem that involves finding the right weights so that the output has the right statistic. The benchmarking between the four methods shows that both DS and WF are comparably accurate to both MC and MLE in most cases tested in this thesis. It was also found that both DS and WF uses approximately the same amount of function calls as MLE and all three methods use a lot fewer function calls to the simulation than MC. It was discovered that WF is not always able to find a solution. This is probably because the methods used for WF are not the optimal method for what they are supposed to do. Finding more optimal methods for WF is something that could be investigated further.